{ "stats": { "total": 2069, "success": 1595, "failed": 474 }, "type_stats": { "ellipse": 467, "hyperbola": 564, "parabola": 532, "circle": 32 }, "reason_counts": { "ellipse_ecc_mismatch": 8, "ellipse_point_off_curve": 8, "hyperbola_point_off_curve": 14, "parabola_point_off_curve": 4, "ellipse_focus_mismatch": 7, "hyperbola_focus_mismatch": 8, "parabola_focus_mismatch": 2 }, "results": [ { "index": 0, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.732385540716972, "b": 2.4497476064802375, "orientation": "horizontal" }, "optimization": { "final_loss": 2.601017381791051e-06, "converged": true, "iterations": 91 }, "output_path": "results/test/hyperbola/problem_0000.png", "answer": "x^2/3-y^2/6=1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G)={F1,F2};Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 1, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0001.png", "answer": "sqrt(10)", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/k^2 = 1);k: Number;k>0;Coordinate(OneOf(Focus(G))) = (3,0)", "coords": {} }, { "index": 2, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0002.png", "answer": "sqrt(34)-1", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 5);PointOnCurve(P,G);Distance(P, yAxis) = d;d:Number", "coords": { "A": [ 4.0, 5.0 ] } }, { "index": 3, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0003.png", "answer": "3*x+4*y-5=0", "fact_expr": "G: Hyperbola;H: LineSegment;M: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(M) = (3, -1);IsChordOf(H,G);MidPoint(H)=M;PointOnCurve(M,H)", "coords": { "M": [ 3.0, -1.0 ] } }, { "index": 4, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0004.png", "answer": "(x^2+y^2=1)&Negation(x=pm*1)", "fact_expr": "P: Point;Slope(LineSegmentOf(P, A))*Slope(LineSegmentOf(P, B)) = -1;A: Point;Coordinate(A) = (-1, 0);B: Point;Coordinate(B) = (1, 0)", "coords": {} }, { "index": 5, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0005.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 6, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0006.png", "answer": "(0,sqrt(2)/2]", "fact_expr": "P: Point;PointOnCurve(P, G) = True;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) <= ApplyUnit(90, degree)", "coords": {} }, { "index": 7, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9872854805122186, "b": 2.6119589473247777, "orientation": "horizontal" }, "optimization": { "final_loss": 4.271082513482038e-06, "converged": true, "iterations": 115 }, "output_path": "results/test/hyperbola/problem_0007.png", "answer": "x^2/2-y^2/14=1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(C) = Focus(G);Expression(OneOf(Asymptote(C))) = (y = sqrt(7)*x)", "coords": {} }, { "index": 8, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_0008.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "H: Ellipse;Expression(H) = (y^2/n + x^2/m = 1);m: Number;n: Number;m>0;n>0;Eccentricity(H) = 1/2;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G)", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 9, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0009.png", "answer": "y^2=-8*x", "fact_expr": "G: Parabola;P: Point;O: Origin;m:Number;Coordinate(P) = (-3, m);Vertex(G) = O;PointOnCurve(Focus(G), xAxis);PointOnCurve(P, G);Distance(P, Focus(G))=5", "coords": {} }, { "index": 10, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0010.png", "answer": "6", "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (1, 5);PointOnCurve(M, G);Focus(G) = F", "coords": { "A": [ 1.0, 5.0 ] } }, { "index": 11, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0011.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 12, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0012.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Line;Expression(G) = (y = b*sqrt(6)/3);Intersection(G, C) = {A, B};A: Point;B: Point;O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True", "coords": {} }, { "index": 13, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0013.png", "answer": "(-1, 1)+{17/8}", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2);H: Circle;Expression(H) = (x^2 + (-a + y)^2 = 1);a: Number;NumIntersection(G, H) = 2", "coords": {} }, { "index": 14, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "y", "x_coef": 5.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0014.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/5 + y^2/9 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);x1:Number;y1:Number", "coords": {} }, { "index": 15, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0015.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;LeftFocus(G) = F1;O: Origin;P: Point;PointOnCurve(P, G);Q: Point;PointOnCurve(Q, RightDirectrix(G));VectorOf(P, Q) = 2*VectorOf(F1, O);lambda: Number;lambda>0;VectorOf(F1, Q) = lambda*(VectorOf(F1, O)/Abs(VectorOf(F1, O)) + VectorOf(F1, P)/Abs(VectorOf(F1, P)))", "coords": {} }, { "index": 16, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0016.png", "answer": "2*sqrt(6)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/b^2 = 1);b: Number;b>0;F: Point;RightFocus(G) = F;Distance(F, OneOf(Asymptote(G))) = 1;H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;p>0;PointOnCurve(LeftFocus(G), Directrix(H)) = True;M: Point;PointOnCurve(M, H) = True;I: Point;Coordinate(I) = (5, 0)", "coords": { "I": [ 5.0, 0.0 ] } }, { "index": 17, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0017.png", "answer": "3", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;E: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);Intersection(H, G) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = E;Abs(LineSegmentOf(E, F)) = 4", "coords": {} }, { "index": 18, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0018.png", "answer": "-2*p/t", "fact_expr": "G: Parabola;p: Number;H: Line;O: Origin;A: Point;M: Point;B: Point;t:Number;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(M,SymmetryAxis(G));Coordinate(M) = (t, 0);PointOnCurve(M, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 19, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.656854249492381, "major_axis": "x", "x_coef": 36.0, "y_coef": 32.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0019.png", "answer": "[21/2, 27/2]", "fact_expr": "G: Ellipse;H: Circle;J: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/36 + y^2/32 = 1);Expression(H) = (y^2 + (x - 2)^2 = 1);Expression(J) = (y^2 + (x + 2)^2 = 1/4);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, J)", "coords": {} }, { "index": 20, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0020.png", "answer": "5", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;A: Point;B: Point;k: Number;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (-1, -1);Slope(l)=k;PointOnCurve(Focus(C),l);Intersection(l, C) = {A,B};PointOnCurve(M,Directrix(C));DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -1.0, -1.0 ] } }, { "index": 21, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.6935477682080864, "b": 1.5214863254929487, "major_axis": "y", "x_coef": 0.9999999999999996, "y_coef": 4.0 }, "optimization": { "final_loss": 1.5072050880173891e-05, "converged": true, "iterations": 98 }, "output_path": "results/test/ellipse/problem_0021.png", "answer": "x^2+y^2/4=1", "fact_expr": "G: Ellipse;A: Point;B: Point;Coordinate(A) = (0, 2);Coordinate(B) = (1/2, sqrt(3));PointOnCurve(A,G) = True;PointOnCurve(B,G) = True", "coords": { "A": [ 0.0, 2.0 ], "B": [ 0.5, 1.7320508075688772 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 22, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0022.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;m: Number;O: Origin;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (m - m*y + x = 0);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Area(TriangleOf(O,A,B))=2*sqrt(2)", "coords": {} }, { "index": 23, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0023.png", "answer": "3/5", "fact_expr": "G: Parabola;P: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P,G);L:Line;PointOnCurve(P,L);IsPerpendicular(L,Directrix(G));FootPoint(L,Directrix(G))=M;Abs(LineSegmentOf(P, M)) = 5;Focus(G) = F", "coords": {} }, { "index": 24, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 25, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0025.png", "answer": "20", "fact_expr": "F2: Point;F1: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);H: Line;PointOnCurve(F1, H) = True;A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 26, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "up" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/test/parabola/problem_0026.png", "answer": "(0,1)", "fact_expr": "G: Parabola;p: Number;H: Point;Expression(G) = (x^2 = 2*(p*y));Coordinate(H) = (2, 1);PointOnCurve(H, G)", "coords": { "H": [ 2.0, 1.0 ] } }, { "index": 27, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0027.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;m>0;Expression(G) = (-x^2 + y^2/m = 1);Expression(H) = (y = x^2/8);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 28, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0028.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);F: Point;LeftFocus(C) = F;l: Line;Slope(l) = sqrt(2);PointOnCurve(F, l);A: Point;B: Point;Intersection(l, LeftPart(C)) = {A, B};E: Circle;IsDiameter(LineSegmentOf(A, B), E);O: Origin;PointOnCurve(O, E)", "coords": {} }, { "index": 29, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 30, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0030.png", "answer": "6/5", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};VectorOf(A, B) = 5*(VectorOf(F, B))", "coords": {} }, { "index": 31, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0031.png", "answer": "{1,3}", "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x = p/2);Focus(G)=F;PointOnCurve(A,G);PointOnCurve(B,G);Abs(VectorOf(A, F)) + Abs(VectorOf(B, F)) = 4;Distance(MidPoint(LineSegmentOf(A,B)),H)=1", "coords": {} }, { "index": 32, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0032.png", "answer": "y=pm*(3/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 33, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0033.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 34, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 35, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0035.png", "answer": "-1", "fact_expr": "k: Number;l: Line;Slope(l) = k;C: Ellipse;Expression(C) = (x^2/3 + y^2/2 = 1);A: Point;B: Point;Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (1/2, 1/3)", "coords": {} }, { "index": 36, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0036.png", "answer": "(-oo,-sqrt(2)/2]+[sqrt(2)/2,+oo)", "fact_expr": "G: Ellipse;H: Line;B: Point;A: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (x = -1/2);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(MidPoint(LineSegmentOf(A,B)),H)", "coords": {} }, { "index": 37, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_0037.png", "answer": "y=pm*2*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 38, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0038.png", "answer": "(1,3]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1,F2};PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 39, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 2.309742261840055, "b": 1.632524068785969, "orientation": "horizontal" }, "optimization": { "final_loss": 4.273270740250148e-06, "converged": true, "iterations": 84 }, "output_path": "results/test/hyperbola/problem_0039.png", "answer": "4*sqrt(3)", "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (4, 2);Expression(Asymptote(C)) = (y = pm*(sqrt(2)/2)*X);PointOnCurve(G, C);PointOnCurve(Focus(C), axis)", "coords": { "G": [ 4.0, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 40, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_0040.png", "answer": "{5/3,5/4}", "fact_expr": "C: Hyperbola;O:Origin;Center(C)=O;Expression(OneOf(Asymptote(C)))=(y=4/3*x);SymmetryAxis(C)=axis", "coords": {} }, { "index": 41, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0041.png", "answer": "1/3", "fact_expr": "E: Parabola;p: Number;C: Circle;G: Line;A: Point;F: Point;P: Point;M: Point;N: Point;p>0;Expression(E) = (x^2 = 2*(p*y));Expression(C) = ((x + 2)^2 + (y + 2)^2 = 1);Coordinate(P) = (3/2, -1);PointOnCurve(P, Directrix(E));TangentOfPoint(P , E) = G;TangentPoint(G, E) = A;Quadrant(A) = 1;Focus(E) = F;PointOnCurve(M, LineOf(A, F));PointOnCurve(N, C)", "coords": { "P": [ 1.5, -1.0 ] } }, { "index": 42, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0042.png", "answer": "3", "fact_expr": "l: Line;C: Parabola;p: Number;p>0;Expression(C) = (y^2 = 2*p*x);Expression(l)=(x-2*y+6=0);IsTangent(l,C)", "coords": {} }, { "index": 43, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0043.png", "answer": "(7+2*sqrt(3))/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F1: Point;P: Point;F2: Point;e:Number;L:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = -x*b/a);LeftFocus(G)=F1;RightFocus(G)=F2;Eccentricity(G)=e;PointOnCurve(F1,L);IsPerpendicular(L,H);Intersection(L,RightPart(G))=P;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 44, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0044.png", "answer": "[sqrt(3)/3,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightDirectrix(G)) = True;P: Point;PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1))) = True", "coords": {} }, { "index": 45, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0045.png", "answer": "3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;k1: Number;k2:Number;A: Point;P: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = k*x);Intersection(G, C) = {A, B};PointOnCurve(P, C);Negation(P = A);Negation(P = B);Slope(LineOf(P,A)) = k1;Slope(LineOf(P,B))=k2;Eccentricity(C) = 2;k:Number", "coords": {} }, { "index": 46, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 47, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0047.png", "answer": "(-3/2,pm*3*sqrt(3))", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;D: Point;Expression(C) = (y^2 = 6*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l,Directrix(C))=D;Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 48, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 49, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0049.png", "answer": "a^2+1+1/(a^2-1)", "fact_expr": "G: Ellipse;a: Number;A: Point;B: Point;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);Eccentricity(G)>sqrt(2)/2;UpperVertex(G)=A;PointOnCurve(B, G)", "coords": {} }, { "index": 50, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 51, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0051.png", "answer": "16/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) = 4", "coords": {} }, { "index": 52, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0052.png", "answer": "[sqrt(2)/2,1]", "fact_expr": "C: Parabola;M: Point;A: Point;F: Point;Expression(C) = (x^2 = 3*y);Focus(C)=F;Intersection(Directrix(C),yAxis)=A;PointOnCurve(M, C)", "coords": {} }, { "index": 53, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0053.png", "answer": "(sqrt(5)+1)/2", "fact_expr": "C1: Hyperbola;Expression(C1)=(x^2/a^2-y^2/b^2=1);a:Number;b:Number;a>0;b>0;F1:Point;LeftFocus(C1)=F1;H:Circle;M:Point;l:Line;Expression(H)=(x^2+y^2=a^2);TangentOfPoint(F1,H)=l;TangentPoint(l,H)=M;C2:Parabola;Expression(C2)=(y^2=2*p*x);p:Number;p>0;N:Point;Intersection(OverlappingLine(LineSegmentOf(F1,M)),C2)=N;OneOf(Focus(C1))=Focus(C2);Abs(LineSegmentOf(M, F1)) = Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 54, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 3.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0054.png", "answer": "ApplyUnit(120, degree)", "fact_expr": "G: Ellipse;B1: Point;F1: Point;B2: Point;Expression(G) = (x^2/12 + y^2/9 = 1);Endpoint(MinorAxis(G)) = {B1, B2};OneOf(Focus(G)) = F1", "coords": {} }, { "index": 55, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0055.png", "answer": "3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = 2*pi;x1: Number;x2: Number;y1: Number;y2: Number;A: Point;B: Point;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": {} }, { "index": 56, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0056.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G);F: Point;Focus(G) = F;Distance(M, F) = 4", "coords": {} }, { "index": 57, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0057.png", "answer": "y = pm*2*sqrt(2)*(x - 1)", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B));LineSegmentOf(A, F) = 2*LineSegmentOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 58, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0058.png", "answer": "5", "fact_expr": "P: Point;Coordinate(P) = (-1, 0);H: Line;PointOnCurve(P, H);G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(H, G) = {A, B};2*Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(A, B))", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 59, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0059.png", "answer": "sqrt(7)/2", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = x);PointOnCurve(B,G);Coordinate(A)=(2,0)", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 60, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0060.png", "answer": "sqrt(3) - 1", "fact_expr": "C: Ellipse;b: Number;a: Number;l1: Line;l2: Line;F1: Point;P: Point;F2: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l1);Inclination(l1) = ApplyUnit(30, degree);PointOnCurve(F2, l2);Intersection(l1, l2) = P;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(90, degree);Eccentricity(C) = e", "coords": {} }, { "index": 61, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 62, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0062.png", "answer": "2", "fact_expr": "G: Hyperbola;A: Point;C:Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (-3, 2*sqrt(3));Asymptote(G)=Asymptote(C);PointOnCurve(A,C)", "coords": { "A": [ -3.0, 3.4641016151377544 ] } }, { "index": 63, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.5129417181015015, "b": 2.0152759552001953, "orientation": "horizontal" }, "optimization": { "final_loss": 3.66731001122389e-06, "converged": true, "iterations": 100 }, "output_path": "results/test/hyperbola/problem_0063.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*(4/3)*x)", "coords": {} }, { "index": 64, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0064.png", "answer": "8", "fact_expr": "G: Parabola;p: Number;M: Circle;A: Point;B:Point;F: Point;x0:Number;y0:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(M) = (y^2 + (x + 2)^2 = 4);Coordinate(A) = (x0, y0);PointOnCurve(A,G);TangentOnPoint(A,G)=l;TangentPoint(l,M)=B;Focus(G)=F;Distance(F,A)=Abs(LineSegmentOf(A,F));l:Line", "coords": {} }, { "index": 65, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0065.png", "answer": "(0, -2)", "fact_expr": "C: Parabola;Expression(C) = (y = -x^2/8)", "coords": {} }, { "index": 66, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0066.png", "answer": "2*x-y-15=0", "fact_expr": "G: Hyperbola;H: Line;B: Point;A: Point;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (8, 1);PointOnCurve(P,H);Intersection(H,G) = {A, B};MidPoint(LineSegmentOf(A,B))=P", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 67, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 68, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0068.png", "answer": "3/4", "fact_expr": "C: Parabola;G: Line;l: Line;A: Point;O: Origin;F: Point;B: Point;D: Point;Expression(C) = (y^2 = 4*x);Directrix(C) = l;Focus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Intersection(LineOf(A,O), l) = D", "coords": {} }, { "index": 69, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0069.png", "answer": "(4*sqrt(3))/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F1, L);A: Point;B: Point;Intersection(L, LeftPart(G)) = {A, B};AngleOf(A, F2, B) = ApplyUnit(60, degree)", "coords": {} }, { "index": 70, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0070.png", "answer": "5/4", "fact_expr": "Length(LineSegmentOf(A, B)) = 3;A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};PointOnCurve(A, G);PointOnCurve(B, G);G: Parabola;Expression(G) = (y^2 = x);M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 71, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 72, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0072.png", "answer": "3*sqrt(2)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;b: Number;a: Number;a>0;b>0;A: Point;H: Line;Expression(H) = (x = (4/3)*a);OneOf(Intersection(H,C)) = A;G: Circle;IsDiameter(LineSegmentOf(F1, F2), G);PointOnCurve(A, G)", "coords": {} }, { "index": 73, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_0073.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (m*y^2 + x^2 = 1);m: Number;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 74, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0074.png", "answer": "{y=((sqrt(3)/3)*x)-sqrt(3)/3, y=((-sqrt(3)/3)*x)+sqrt(3)/3}", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;Q: Point;Projection(MidPoint(LineSegmentOf(A, F)), yAxis) = P;Projection(MidPoint(LineSegmentOf(B, F)), yAxis) = Q;Abs(LineSegmentOf(P, Q)) = 4", "coords": {} }, { "index": 75, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0075.png", "answer": "x^2-y^2/3=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);PointOnCurve(OneOf(Focus(G)), Directrix(H));Eccentricity(G) = 2", "coords": {} }, { "index": 76, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0076.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Expression(H) = (x^2 + y^2 = a^2);M: Point;AngleOf(F1, M, F2) = pi/4;L:Line;TangentOfPoint(F1,H)=L;Intersection(L,RightPart(G))=M", "coords": {} }, { "index": 77, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0077.png", "answer": "sqrt(2)", "fact_expr": "A: Point;Coordinate(A) = (0, 2);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;B: Point;Intersection(LineSegmentOf(F, A), G) = B;Z: Line;PointOnCurve(B, Z);IsPerpendicular(Z, l);M: Point;FootPoint(Z, l) = M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 78, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0078.png", "answer": "(sqrt(13) + 1)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;O: Origin;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;IsDiameter(LineSegmentOf(O, F), H);OneOf(Intersection(H, G)) = A;Inclination(LineSegmentOf(O, A)) = pi/6", "coords": {} }, { "index": 79, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0079.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);H: Line;PointOnCurve(RightFocus(G), H);IsPerpendicular(H, xAxis);A: Point;B: Point;L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(H, L1) = A;Intersection(H, L2) = B", "coords": {} }, { "index": 80, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0080.png", "answer": "b^2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;M: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Negation(P=Vertex(G));LeftFocus(G) = F1;RightFocus(G) = F2;HalfFocalLength(G) = c;c:Number;TangentPoint(InscribedCircle(TriangleOf(P,F1,F2)),LineSegmentOf(F1,F2))=M", "coords": {} }, { "index": 81, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 0.9927864074707031, "direction": "right" }, "optimization": { "final_loss": 0.00015326803259085864, "converged": true, "iterations": 44 }, "output_path": "results/test/parabola/problem_0081.png", "answer": "(125*pi)/16", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;P: Point;Coordinate(P) = (4, 4);PointOnCurve(P, C);F: Point;Focus(C) = F;M: Point;Coordinate(M) = (-1, 4)", "coords": { "P": [ 4.0, 4.0 ], "M": [ -1.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 82, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0082.png", "answer": "4*x - y - 15 = 0", "fact_expr": "G: Parabola;A: Point;B: Point;Q: Point;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A,B),G);Coordinate(Q) = (4, 1);MidPoint(LineSegmentOf(A,B))=Q", "coords": { "Q": [ 4.0, 1.0 ] } }, { "index": 83, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0083.png", "answer": "x=-3/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;P: Point;F: Point;Q: Point;O: Origin;p>0;Focus(C)=F;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);PointOnCurve(Q, xAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P));Abs(LineSegmentOf(F, Q)) = 6", "coords": {} }, { "index": 84, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0084.png", "answer": "x^2/4-y^2/12=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-a + x)^2 = 3);Eccentricity(C) = 2;IsTangent(Asymptote(C),G)", "coords": {} }, { "index": 85, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0085.png", "answer": "{(sqrt(3)/2, +oo), (-oo, -sqrt(3)/2)}", "fact_expr": "l: Line;O: Origin;PointOnCurve(O, l) = True;G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = -1);NumIntersection(l, G) = 2", "coords": {} }, { "index": 86, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0086.png", "answer": "2", "fact_expr": "M: Point;Coordinate(M) = (-1, 1);C: Parabola;Expression(C) = (y^2 = 4*x);G: Line;PointOnCurve(Focus(C),G) = True;Slope(G) = k;k: Number;Intersection(G, C) = {A, B};A: Point;B: Point;AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 87, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0087.png", "answer": "", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;m>0;Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;G: Hyperbola;Expression(G) = (-y^2 + x^2/a = 1);a: Real;A: Point;LeftVertex(G) = A;IsParallel(OneOf(Asymptote(G)), LineOf(A, M))", "coords": {} }, { "index": 88, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 89, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0089.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Expression(Asymptote(G)) = (sqrt(3)*x + pm*y = 0)", "coords": {} }, { "index": 90, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 91, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0091.png", "answer": "4", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 1);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 92, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 93, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0093.png", "answer": "(3*sqrt(3)/4, +oo)", "fact_expr": "C: Hyperbola;M: Point;F1: Point;F2: Point;P: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, LeftPart(C));Perimeter(TriangleOf(M, F1, F2)) = 9;PointOnCurve(P, RightPart(C))", "coords": {} }, { "index": 94, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 95, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 96, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0096.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;O: Origin;A: Point;F1: Point;F2: Point;x0: Number;y0: Number;Expression(G) = (x^2/9 + y^2/8 = 1);Coordinate(P) = (x0, y0);LeftFocus(G) = F1;RightFocus(G) = F2;RightVertex(G) = A;DotProduct(VectorOf(O, P), VectorOf(O, A)) = 9;WhenMax(AngleOf(F1, P, F2))", "coords": {} }, { "index": 97, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0097.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;A: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;Abs(LineSegmentOf(F1,F2))=2*c;O:Origin;Center(A)=O;Radius(A)=c;OneOf(Intersection(A, C)) = P;Area(TriangleOf(F1,P,F2)) = a^2", "coords": {} }, { "index": 98, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0098.png", "answer": "2*sqrt(10)", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 99, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 0.8176298651812968, "b": 2.452642014841466, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4992967561765154e-06, "converged": true, "iterations": 283 }, "output_path": "results/test/hyperbola/problem_0099.png", "answer": "y^2/9 - x^2 = 1", "fact_expr": "E: Hyperbola;F: Point;Coordinate(F) = (-sqrt(3), 6);PointOnCurve(F, E);Expression(Asymptote(E)) = (y = pm*3*x)", "coords": { "F": [ -1.7320508075688772, 6.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 100, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 101, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0101.png", "answer": "5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);P: Point;t: Number;Coordinate(P) = (t, 3);PointOnCurve(P, C);F: Point;Focus(C) = F", "coords": {} }, { "index": 102, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 103, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0103.png", "answer": "4", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/9 - y^2/b^2 = 1);Distance(RightFocus(G),Asymptote(G))=(1/2)*Distance(RightFocus(G),LeftVertex(G))", "coords": {} }, { "index": 104, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 105, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 106, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 107, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0107.png", "answer": "x^2/9 + y^2/4 = 1", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(F1, F2));Abs(LineSegmentOf(P, F1)) = 4/3;Abs(LineSegmentOf(P, F2)) = 14/3", "coords": {} }, { "index": 108, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0108.png", "answer": "14+8*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 8);RightFocus(G) = F2;F2: Point;P: Point;Q: Point;PointOnCurve(F2, LineSegmentOf(P, Q)) = True;IsChordOf(LineSegmentOf(P, Q), G) = True;Abs(LineSegmentOf(P, Q)) = 7;F1: Point;LeftFocus(G) = F1", "coords": {} }, { "index": 109, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7994307279586792, "b": 1.695713604329748, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.1180587689958606e-05, "converged": true, "iterations": 86 }, "output_path": "results/test/ellipse/problem_0109.png", "answer": "4", "fact_expr": "G: Ellipse;I: Point;P: Point;F1: Point;F2: Point;Q: Point;lambda:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);Quadrant(P)=1;Incenter(TriangleOf(P, F1, F2)) = I;Intersection(LineOf(P,I), xAxis) = Q;Eccentricity(G) = 1/3;VectorOf(P, Q) = lambda*VectorOf(I, Q);a:Number;b:Number;a>b;b>0", "coords": {} }, { "index": 110, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0110.png", "answer": "pm*sqrt(5)/2", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};VectorOf(A, F) = 5*VectorOf(F, B)", "coords": {} }, { "index": 111, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0111.png", "answer": "(x-5)^2+y^2=16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);H: Circle;RightFocus(G) = Center(H);IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 112, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0112.png", "answer": "sqrt(6)/2", "fact_expr": "C: Ellipse;a: Real;a>1;Expression(C) = (y^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)/3", "coords": {} }, { "index": 113, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0113.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 114, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 115, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0115.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;L: Line;PointOnCurve(F, L) ;IsPerpendicular(L, OneOf(Asymptote(G))) ;Asymptote(G) = {l1, l2};Intersection(L, l1) = A;Intersection(L, l2) = B;A: Point;B: Point;l1: Line;l2: Line;Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(B, F)) = 1/2", "coords": {} }, { "index": 116, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0116.png", "answer": "(0, pm*b)", "fact_expr": "G: Ellipse;b: Number;a: Number;F: Point;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);OneOf(Focus(G)) = F;P: Point;PointOnCurve(P, G);Max(Distance(F, P)) = m ;m: Number;n: Number;Min(Distance(F, P)) = n;P0: Point;PointOnCurve(P0, G);Distance(F, P0) = (m + n)/2", "coords": {} }, { "index": 117, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0117.png", "answer": "(x^2/4 + y^2/3 = 1)&Negation(y = 0)", "fact_expr": "G: Parabola;H: Circle;A: Point;B: Point;Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);Expression(H) = (x^2 + y^2 = 4);PointOnCurve(A, G);PointOnCurve(B, G);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 118, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0118.png", "answer": "{sqrt(6)/2, -sqrt(6)/2, sqrt(2)/2, -sqrt(2)/2}", "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Real;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);NumIntersection(H, G) = 1", "coords": {} }, { "index": 119, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0119.png", "answer": "3, 25/3", "fact_expr": "G: Ellipse;m: Number;e: Number;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = e ;e = sqrt(10)/5", "coords": {} }, { "index": 120, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0120.png", "answer": "(-oo,-1]+[3,+oo)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);B: Point;Coordinate(B) = (1, 1);PointOnCurve(B, G) = True;PointOnCurve(P, G) = True;PointOnCurve(Q, G) = True;P: Point;Q: Point;IsPerpendicular(LineSegmentOf(B, P), LineSegmentOf(P, Q)) = True", "coords": { "B": [ 1.0, 1.0 ] } }, { "index": 121, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0121.png", "answer": "(0,-2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)", "coords": {} }, { "index": 122, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0122.png", "answer": "5*sqrt(2)/2-1", "fact_expr": "l: Line;G: Parabola;P: Point;d1:Number;d2:Number;Expression(G) = (y^2 = 4*x);Expression(l) = (x - y + 4 = 0);PointOnCurve(P, G);Distance(P, yAxis) = d1;Distance(P, l) = d2", "coords": {} }, { "index": 123, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0123.png", "answer": "", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 124, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0124.png", "answer": "1", "fact_expr": "C: Parabola;E: Point;M: Point;N: Point;lambda: Real;lambda>=0;Expression(C) = (y^2 = 4*x);Coordinate(E) = (-lambda, 0);PointOnCurve(M, C);PointOnCurve(N, C);Min(DotProduct(VectorOf(E, M), VectorOf(E, N))) = 0", "coords": {} }, { "index": 125, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0125.png", "answer": "3", "fact_expr": "C: Parabola;F: Point;N: Point;M: Point;l: Line;lambda: Number;lambda > 0;Expression(C) = (y^2 = 12*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(N, l);LineSegmentOf(F, N) = lambda*LineSegmentOf(F, M);Abs(LineSegmentOf(M, F)) = 4", "coords": {} }, { "index": 126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 127, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 128, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0128.png", "answer": "(pm*3,0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/25 = 1);P:Point;F1:Point;F2:Point;PointOnCurve(P,G);Focus(G)={F1,F2};WhenMax(Distance(P,F1)*Distance(P,F2))", "coords": {} }, { "index": 129, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0129.png", "answer": "y=pm*(3/5)*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/25 - y^2/9 = 1)", "coords": {} }, { "index": 130, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0130.png", "answer": "6", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);M: Point;PointOnCurve(M, C) = True;F1: Point;F2: Point;Focus(C) = {F1, F2}", "coords": {} }, { "index": 131, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0131.png", "answer": "y^2=16*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Vertex(G) = Center(H);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 132, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0132.png", "answer": "10", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4", "coords": {} }, { "index": 133, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 134, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 135, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 136, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0136.png", "answer": "p=1/2", "fact_expr": "C: Parabola;p: Number;A: Point;p>0;Expression(C) = (y^2 = 2*p*x);m:Number;Coordinate(A) = (4, m);PointOnCurve(A, C);Distance(A, Focus(C)) = 17/4", "coords": {} }, { "index": 137, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0137.png", "answer": "4*sqrt(10)/5", "fact_expr": "G: Ellipse;H: Line;t: Number;A: Point;B: Point;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = t + x);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 138, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0138.png", "answer": "pm*4*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);O: Origin;N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;Abs(LineSegmentOf(F, N)) = 2*Abs(LineSegmentOf(O, M))", "coords": {} }, { "index": 139, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.8706156235768385, "b": 0.9134866531697662, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 8.509210183089842e-07, "converged": true, "iterations": 93 }, "output_path": "results/test/ellipse/problem_0139.png", "answer": "x^2/9+y^2=1", "fact_expr": "G: Ellipse;P: Point;Coordinate(P) = (3, 0);PointOnCurve(Focus(G),xAxis);PointOnCurve(P,G);Length(MajorAxis(G))=3*Length(MinorAxis(G))", "coords": { "P": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 140, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 141, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0141.png", "answer": "(-oo, -sqrt(2)]", "fact_expr": "P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;G: Hyperbola;Expression(G) = (x^2/2 - y^2/4 = 1);PointOnCurve(P, LeftPart(G))", "coords": {} }, { "index": 142, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0142.png", "answer": "16", "fact_expr": "l: Line;PointOnCurve(F2, l);G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;Intersection(l, G) = {A, B}", "coords": {} }, { "index": 143, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 144, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0144.png", "answer": "5/8", "fact_expr": "l: Line;C: Parabola;p: Number;G: Circle;B: Point;A: Point;Q: Point;M: Point;N: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Slope(l) = sqrt(3);Intersection(l, C) = {A, B};IsDiameter(LineSegmentOf(A, B), G);Intersection(G, yAxis) = {M, N};MidPoint(LineSegmentOf(A, B)) = Q;Distance(F, Directrix(C)) = 3", "coords": {} }, { "index": 145, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0145.png", "answer": "y = pm*x*8/15", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;A: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;Slope(G) = 1/4;PointOnCurve(F, G);Intersection(G, OneOf(Asymptote(C))) = A;Quadrant(A) = 1;Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 146, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0146.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (16*x^2 - 9*y^2 = 144)", "coords": {} }, { "index": 147, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 148, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 149, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0149.png", "answer": "3/2", "fact_expr": "l: Line;G: Parabola;B: Point;C: Point;F: Point;A: Point;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};Intersection(l,Directrix(G))=C;Focus(G) = F;VectorOf(B, C) = 3*VectorOf(F, B)", "coords": {} }, { "index": 150, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0150.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;A: Point;l:Line;B: Point;C: Point;D: Point;Expression(G) = (x^2/3 + y^2/4 = 1);Expression(H) = (x + y + 1 = 0);Expression(l) = (x+y-1=0);UpperFocus(G) = F;Intersection(H,G)={A,B};Intersection(l,G)={C,D};F:Point", "coords": {} }, { "index": 151, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0151.png", "answer": "", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/9 = 1);P: Point;Coordinate(P) = (1/2, 1/2);L: Line;PointOnCurve(P,L);Intersection(P,G) = {A,B};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 0.5, 0.5 ] } }, { "index": 152, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.4142135623730951 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0152.png", "answer": "{5, 1/3}", "fact_expr": "l: Line;C: Circle;A: Point;B: Point;P: Point;Expression(C) = (y^2 + (x - 2)^2 = 2);Expression(l) = (x - y + 2 = 0);Intersection(l, xAxis) = A;PointOnCurve(P, l);PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, P))", "coords": {} }, { "index": 153, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 154, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 155, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0155.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + y^2 = a^2);L1: Line;L2: Line;TangentOfPoint(OneOf(Focus(C)), G) = {L1, L2};A: Point;B: Point;TangentPoint(L1, G) = A;TangentPoint(L2, G) = B;AngleOf(A, O, B) = ApplyUnit(120, degree);O: Origin", "coords": {} }, { "index": 156, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0156.png", "answer": "1", "fact_expr": "P: Point;Q: Point;PointOnCurve(P,G) = True;PointOnCurve(Q,G) = True;G: Parabola;Expression(G) = (x^2 = 2*y);XCoordinate(P) = 4;XCoordinate(Q) = -2;L1: Line;L2: Line;TangentOfPoint(P,G) = L1;TangentOfPoint(Q,G) = L2;Intersection(L1,L2) = A;A:Point", "coords": {} }, { "index": 157, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0157.png", "answer": "(sqrt(17)-3)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(M, G);Distance(M,LeftFocus(G))=2*Distance(M,RightDirectrix(G))", "coords": {} }, { "index": 158, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0158.png", "answer": "x^2/16-y^2/9=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;C:Parabola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (4, 3);PointOnCurve(H,OneOf(Asymptote(G)));Expression(C)=(y^2=20*x);PointOnCurve(OneOf(Focus(G)),Directrix(C))", "coords": { "H": [ 4.0, 3.0 ] } }, { "index": 159, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0159.png", "answer": "(0, 1]+[9, +oo)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/m = 1);m: Number;A: Point;B: Point;Endpoint(MajorAxis(C)) = {A, B};M: Point;PointOnCurve(M, C);AngleOf(A, M, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 160, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0160.png", "answer": "2/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1)", "coords": {} }, { "index": 161, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0161.png", "answer": "(1, 2]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 162, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0162.png", "answer": "9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 163, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0163.png", "answer": "x^2/10 - y^2/6 = 1", "fact_expr": "G: Hyperbola;Z: Hyperbola;Expression(G) = (x^2/5 - y^2/3 = 1);Asymptote(Z) = Asymptote(G);Expression(OneOf(Directrix(Z))) = (x = 5/2)", "coords": {} }, { "index": 164, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0164.png", "answer": "8", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Inclination(LineSegmentOf(A, F)) = (2*pi)/3", "coords": {} }, { "index": 165, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0165.png", "answer": "x^2+y^2=4/3", "fact_expr": "G: Circle;Expression(G) = (x^2 + y^2 = 1);P: Point;TangentOfPoint(P,G) = {LineSegmentOf(P,A),LineSegmentOf(P,B)};TangentPoint(LineSegmentOf(P,A),G) = A;TangentPoint(LineSegmentOf(P,B),G) = B;A: Point;B: Point;AngleOf(A, P, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 166, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0166.png", "answer": "16", "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/8 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l) ;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 167, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0167.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, C);O: Origin;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F2));Tan(AngleOf(P, F2, F1)) = 3", "coords": {} }, { "index": 168, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 169, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0169.png", "answer": "5/7", "fact_expr": "l: Line;C: Ellipse;A: Point;F2: Point;F1: Point;B: Point;LeftFocus(C) = F1;RightFocus(C)=F2;Expression(C)=(x^2/a^2 + y^2/b^2 = 1);PointOnCurve(F1, l);Intersection(l, C) = {A, B};DotProduct((VectorOf(A,F2)+VectorOf(F1,F2)),VectorOf(A,F1))=0;VectorOf(A,F1)=4/3*VectorOf(F1,B);a:Number;b:Number;a>b;b>0", "coords": {} }, { "index": 170, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0170.png", "answer": "y = -1/8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/2)", "coords": {} }, { "index": 171, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0171.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);O: Origin;Abs(VectorOf(O, P)) = Abs(VectorOf(O, F1));Area(TriangleOf(F1, F2, P)) = 4", "coords": {} }, { "index": 172, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0172.png", "answer": "y^2=6*x", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = sqrt(3);Intersection(l, C) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A,B))) = sqrt(3)", "coords": {} }, { "index": 173, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0173.png", "answer": "sqrt(15)", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 1);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 174, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0174.png", "answer": "16/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 175, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 4.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0175.png", "answer": "x^2/48-y^2/16=1", "fact_expr": "G: Hyperbola;H: Ellipse;O:Origin;Expression(H) = (x^2 + 4*y^2 = 64);Center(G)=O;Vertex(G)=Focus(H);FocalLength(G) = 16", "coords": {} }, { "index": 176, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0176.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightFocus(G), Asymptote(G)) = sqrt(3)/4*FocalLength(G)", "coords": {} }, { "index": 177, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0177.png", "answer": "8", "fact_expr": "l: Line;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1,y1);Coordinate(Q) = (x2,y2);PointOnCurve(Focus(G),l);Intersection(l, G) = {P,Q};x1:Number;x2:Number;y1:Number;y2:Number;x1+x2=6;G:Parabola", "coords": {} }, { "index": 178, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 5.656854249492381, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0178.png", "answer": "2", "fact_expr": "G: Hyperbola;A: Point;x0:Number;y0:Number;Expression(G) = (x^2/4 - y^2/32 = 1);Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));Distance(A,RightFocus(G)) = 2*x0", "coords": {} }, { "index": 179, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0179.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = -4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Slope(H) = sqrt(3)", "coords": {} }, { "index": 180, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/test/hyperbola/problem_0180.png", "answer": "x^2/3-y^2/6=1", "fact_expr": "G: Hyperbola;Coordinate(Focus(G)) = (pm*3, 0);Expression(Asymptote(G)) = (y=pm*(sqrt(2)*x))", "coords": {} }, { "index": 181, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0181.png", "answer": "y = 1", "fact_expr": "C: Parabola;Expression(C) = (y = -x^2/4)", "coords": {} }, { "index": 182, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0182.png", "answer": "12", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 4);RightFocus(C)= F;PointOnCurve(P,LeftPart(C))", "coords": { "A": [ 0.0, 4.0 ] } }, { "index": 183, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 184, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.33732376966822, "b": 1.7454107534785184, "orientation": "horizontal" }, "optimization": { "final_loss": 1.8920117601792354e-05, "converged": true, "iterations": 51 }, "output_path": "results/test/hyperbola/problem_0184.png", "answer": "{5/4, 5/3}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/4)*x)", "coords": {} }, { "index": 185, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0185.png", "answer": "y^2/7-x^2/14=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);G1: Hyperbola;Asymptote(G) = Asymptote(G1);H: Point;Coordinate(H) = (-2, -3);PointOnCurve(H, G1) = True", "coords": { "H": [ -2.0, -3.0 ] } }, { "index": 186, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0186.png", "answer": "10", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2}", "coords": {} }, { "index": 187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.3645083904266357, "b": 1.1801031827926636, "orientation": "horizontal" }, "optimization": { "final_loss": 1.282816697312228e-06, "converged": true, "iterations": 88 }, "output_path": "results/test/hyperbola/problem_0187.png", "answer": "5/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;Omega: Circle;M: Point;F1: Point;F2: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Omega) = (-2*y + x^2 + y^2 = 0);LeftFocus(C) = F1;RightFocus(C) = F2;Length(RealAxis(C)) = 2;Expression(Asymptote(C)) = (y = pm*(1/2)*x);Abs(LineSegmentOf(M, F1)) - Abs(LineSegmentOf(M, F2)) = 2;PointOnCurve(N, Omega)", "coords": {} }, { "index": 188, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0188.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Line;O: Origin;A: Point;F:Point;p>0;Expression(G) = (y^2 = p*x);Slope(H)=2;Focus(G)=F;PointOnCurve(F,H);Intersection(H,yAxis)=A;Area(TriangleOf(O,A,F))=1", "coords": {} }, { "index": 189, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 190, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 191, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 192, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0192.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/3 - y^2 = 1);p>0;Expression(H) = (y^2 = -2*p*x);Focus(H) = LeftFocus(G)", "coords": {} }, { "index": 193, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 194, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/test/hyperbola/problem_0194.png", "answer": "y^2/3-x^2=1", "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Point;Coordinate(H) = (0, 2);OneOf(Focus(G)) = H", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 195, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0195.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2=2*p*x);p: Number;p>0;A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;Focus(G) = F;F: Point;PointOnCurve(F,LineSegmentOf(A,B)) = True;l: Line;Directrix(G) = l;IsPerpendicular(LineSegmentOf(A,A1),l) = True;A1: Point;FootPoint(LineSegmentOf(A,A1),l) = A1;IsPerpendicular(LineSegmentOf(B,B1),l) = True;B1: Point;FootPoint(LineSegmentOf(B,B1),l) = B1;N: Point;MidPoint(LineSegmentOf(A1,B1)) = N;LineSegmentOf(A,A1) = 4;LineSegmentOf(B,B1) = 2", "coords": {} }, { "index": 196, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 197, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 198, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 199, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0199.png", "answer": "4*sqrt(3)/3", "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Inclination(l)=pi/3;PointOnCurve(F,l);Intersection(l,C)={A,B}", "coords": {} }, { "index": 200, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 201, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0201.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Distance(P,F) = 3", "coords": {} }, { "index": 202, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0202.png", "answer": "2\nApplyUnit(120, degree)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0203.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2=2*p*x);Coordinate(A) = (2, 4);PointOnCurve(OneOf(Focus(G)), Directrix(H));Eccentricity(G) = 2;p:Number;PointOnCurve(A,H)", "coords": { "A": [ 2.0, 4.0 ] } }, { "index": 204, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 205, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0205.png", "answer": "12", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1, l);P: Point;Q: Point;Intersection(l, LeftPart(G)) = {P, Q};Abs(LineSegmentOf(P, Q)) = 4", "coords": {} }, { "index": 206, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0206.png", "answer": "sqrt(2)/4", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;A: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) ={F1,F2};IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Intersection(LineSegmentOf(P, F1), C) = A;Intersection(LineSegmentOf(P, F2), C) =B;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(A, F1))", "coords": {} }, { "index": 207, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0207.png", "answer": "(-oo,-6)", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8);A: Point;Intersection(Directrix(G), SymmetryAxis(G)) = A;H: Line;PointOnCurve(A, H) = True;Intersection(H, G) = {M, N};M: Point;N: Point;PointOnCurve(B, SymmetryAxis(G)) = True;B: Point;DotProduct((VectorOf(B, M) + VectorOf(M, N)/2), VectorOf(M, N)) = 0;O: Origin", "coords": {} }, { "index": 208, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0208.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(P, F1, F2) = pi/6;Abs(VectorOf(P, F1) + VectorOf(P, F2)) = Abs(VectorOf(F1, F2))", "coords": {} }, { "index": 209, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0209.png", "answer": "{3,6}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;L: Line;PointOnCurve(F,L);Intersection(L, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B));H: Circle;Expression(H) = (x^2 + y^2 = 2*p);Length(InterceptChord(LineOf(A, B), H)) = 4", "coords": {} }, { "index": 210, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0210.png", "answer": "4", "fact_expr": "G: Parabola;H: Circle;F: Point;A: Point;M: Point;O:Origin;Vertex(G)=O;Coordinate(F) = (0, 1);Focus(G)=F;Expression(H) = ((x + 1)^2 + (y - 4)^2 = 1);PointOnCurve(A,G);PointOnCurve(M,H)", "coords": { "F": [ 0.0, 1.0 ] } }, { "index": 211, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0211.png", "answer": "{5,3}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2", "coords": {} }, { "index": 212, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0212.png", "answer": "sqrt(13)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;F: Point;O: Origin;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(F,l);IsPerpendicular(l,OneOf(Asymptote(C)));Intersection(l,RightPart(C))=P;MidPoint(LineSegmentOf(P, F)) = A;Abs(LineSegmentOf(O, A)) = -a + (3*b)/2", "coords": {} }, { "index": 213, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0213.png", "answer": "sqrt(7)/4", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 214, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 215, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 216, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 217, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0217.png", "answer": "1/2", "fact_expr": "G: Parabola;I: Point;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 4*x);Coordinate(I) = (0, 2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);IsChordOf(LineSegmentOf(A, B), G);Intersection(OverlappingLine(LineSegmentOf(A, B)), yAxis) = I", "coords": { "I": [ 0.0, 2.0 ] } }, { "index": 218, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 219, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 220, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0220.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;H: Line;M: Point;A: Point;p>0;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (p, 0);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H,G) = {A, B};y1*y2=-2;B:Point;PointOnCurve(M,H)", "coords": {} }, { "index": 221, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0221.png", "answer": "4*a+2*m", "fact_expr": "A: Point;B: Point;G: Hyperbola;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, RightPart(G));PointOnCurve(B, RightPart(G));Focus(G)={F1,F2};RightFocus(G) = F2;PointOnCurve(F2, LineSegmentOf(A, B));Abs(LineSegmentOf(A, B)) = m;m:Number;b:Number;a:Number", "coords": {} }, { "index": 222, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0222.png", "answer": "0.2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;L:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,L);IsPerpendicular(L,xAxis);Intersection(L,G)=P;AngleOf(F1,P,F2)=ApplyUnit(45,degree)", "coords": {} }, { "index": 223, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0223.png", "answer": "-2", "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;H: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(H) = (2, 0);PointOnCurve(H, l);Intersection(l, C) = {A, B}", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 224, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0224.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 225, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0225.png", "answer": "4*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P,RightPart(C));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Expression(Asymptote(C)) = (y = pm*x);Area(TriangleOf(P, F1, F2)) = 4*sqrt(2)", "coords": {} }, { "index": 226, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 227, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0227.png", "answer": "25*pi/8", "fact_expr": "L: Line;Expression(L) = (x - y - 4 = 0);P: Point;PointOnCurve(P,L) = True;Negation(PointOnCurve(P,xAxis));Z: Parabola;Expression(Z) = (x^2 = 4*y);L1: Line;L2: Line;TangentOfPoint(P,Z) = {L1,L2};Intersection(L1,xAxis) = G;G: Point;Intersection(L2,xAxis) = H;H: Point", "coords": {} }, { "index": 228, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0228.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 229, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0229.png", "answer": "-1/2", "fact_expr": "m: Line;G: Ellipse;O: Origin;P: Point;P1: Point;P2: Point;Expression(G) = (x^2/2 + y^2 = 1);Intersection(m, G) = {P1, P2};MidPoint(LineSegmentOf(P1, P2)) = P;k1: Number;Slope(m) = k1;Negation(k1 = 0);k2: Number;Slope(LineOf(O, P)) = k2", "coords": {} }, { "index": 230, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0230.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/b^2 = 1);b: Number;0 < b;b < 2;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l) = True;l: Line;Intersection(l, G) = {A, B};B: Point;A: Point;Max(Abs(VectorOf(A, F2)) + Abs(VectorOf(B, F2))) = 5", "coords": {} }, { "index": 231, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 232, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0232.png", "answer": "1/2", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y = x*(sqrt(3)/3))", "coords": {} }, { "index": 233, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 234, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 235, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 236, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 237, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch;ellipse_focus_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.9999999999999996 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_0237.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;F: Point;O:Origin;Coordinate(F) = (1, 0);RightFocus(C) = F;Center(C)=O;Eccentricity(C)=1/2", "coords": { "F": [ 1.0, 0.0 ] }, "validation_reasons": [ "ellipse_ecc_mismatch", "ellipse_focus_mismatch" ] }, { "index": 238, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0238.png", "answer": "3", "fact_expr": "G: Parabola;M: Point;F: Point;Focus(G)=F;Expression(G) = (y^2 = 4*x);PointOnCurve(M,G);Distance(M,F)=Abs(LineSegmentOf(M,F));Abs(LineSegmentOf(M,F))=4;XCoordinate(M)=x1;x1:Number", "coords": {} }, { "index": 239, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0239.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (x^2 = 2*(p*y));Expression(H) = (y^2 + (x - 1)^2 = 9);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 240, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0240.png", "answer": "3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 241, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 242, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 243, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 244, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0244.png", "answer": "y^2=-36*(x-5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);H: Parabola;RightFocus(G) = Vertex(H);LeftVertex(G) = Focus(H)", "coords": {} }, { "index": 245, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/test/hyperbola/problem_0245.png", "answer": "{2,2*sqrt(3)/3}", "fact_expr": "G: Hyperbola;O:Origin;Center(G)=O;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 246, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 6.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0246.png", "answer": "12", "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/64 + y^2/36 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 4", "coords": {} }, { "index": 247, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 16.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0247.png", "answer": "16", "fact_expr": "G: Ellipse;B: Point;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/6 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B}", "coords": {} }, { "index": 248, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0248.png", "answer": "[-7, 1]", "fact_expr": "l: Line;G: Ellipse;C: Point;A: Point;B: Point;Expression(G) = (x^2 + y^2/9 = 1);Coordinate(C) = (1, 1);a: Number;b: Number;Expression(l) = (a*x + b*y = 0);Intersection(l, G) = {A, B}", "coords": { "C": [ 1.0, 1.0 ] } }, { "index": 249, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0249.png", "answer": "{0}+[1/2, oo)", "fact_expr": "G: Parabola;H: Line;k: Number;Expression(G) = (y^2 = 2*x);Expression(H) = (y = k*x + 1);NumIntersection(H, G) <= 1", "coords": {} }, { "index": 250, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 251, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 252, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0252.png", "answer": "-4/5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;Expression(G) = (y = 2*x - 4);A: Point;B: Point;Intersection(G, C) = {A, B}", "coords": {} }, { "index": 253, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0253.png", "answer": "sqrt(3)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;B1: Point;B2: Point;UpperVertex(C) = B1;LowerVertex(C) = B2;M: Point;PointOnCurve(M, C);Negation(PointOnCurve(M, axis));Slope(LineOf(M, B1))*Slope(LineOf(M, B2)) = -b/(2*a)", "coords": {} }, { "index": 254, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0254.png", "answer": "{(-oo,0)+[4,oo)}", "fact_expr": "G: Parabola;H: Curve;a: Real;Expression(G) = (y^2 = 8*x);Expression(H) = And(x^2/a+y^2/4=1,y>=0);NumIntersection(Directrix(G), H) = 1", "coords": {} }, { "index": 255, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0255.png", "answer": "2*sqrt(2)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H, G) = {A, B};WhenMin(Abs(LineSegmentOf(A, F)) + 4*Abs(LineSegmentOf(B, F)))", "coords": {} }, { "index": 256, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 257, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0257.png", "answer": "sqrt(2)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;PointOnCurve(A, G);F: Point;OneOf(Focus(G)) = F;IsPerpendicular(LineSegmentOf(A, F), xAxis);FocalLength(G) = Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 258, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0258.png", "answer": "1/2", "fact_expr": "G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = lambda*VectorOf(F, B);Abs(VectorOf(A, F)) = 3/2;lambda:Number", "coords": {} }, { "index": 259, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0259.png", "answer": "32/3", "fact_expr": "G: Parabola;p: Number;H: Line;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(G) = F;PointOnCurve(F, H);Intersection(H, Directrix(G)) = A;OneOf(Intersection(H, G)) = B;VectorOf(F, A) = -3*VectorOf(F, B)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 260, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0260.png", "answer": "9", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F: Point;LeftFocus(C) = F;A: Point;Coordinate(A) = (-3, -3);P: Point;PointOnCurve(P, C)", "coords": { "A": [ -3.0, -3.0 ] } }, { "index": 261, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0261.png", "answer": "2*sqrt(13)", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (y^2 = -8*x);Focus(G) = F;PointOnCurve(P, Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 262, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0262.png", "answer": "sqrt(5)/2", "fact_expr": "G: Parabola;p: Number;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Quadrant(A) = 1;Quadrant(B) = 1;Abs(LineSegmentOf(A, F)) = 2;Abs(LineSegmentOf(B, F)) = 4;Abs(LineSegmentOf(A, B)) = 3", "coords": {} }, { "index": 263, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0263.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G),H);IsPerpendicular(RealAxis(G),H);Intersection(H,G) = {M,N};IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(O, N))", "coords": {} }, { "index": 264, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0264.png", "answer": "9", "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x - 5)^2 = 4);Expression(Z) = (y^2 + (x + 5)^2 = 1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, Z)", "coords": {} }, { "index": 265, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0265.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(2)/2));Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 266, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0266.png", "answer": "x^2-4*y^2=1", "fact_expr": "O: Origin;P: Point;G: Hyperbola;M: Point;Expression(G) = (x^2/4 - y^2 = 1);PointOnCurve(P, G);MidPoint(LineSegmentOf(O, P)) = M", "coords": {} }, { "index": 267, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.3166247903554, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 11.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0267.png", "answer": "4*sqrt(11)", "fact_expr": "l: Line;G: Ellipse;F1: Point;A: Point;B: Point;F2: Point;Expression(G) = (x^2/11 + y^2/7 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 268, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0268.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;c: Number;Coordinate(F) = (c, 0);RightFocus(C) = F;P: Point;PointOnCurve(P, C);G: Circle;Expression(G) = (y^2 + (-c/3 + x)^2 = b^2/9);Q: Point;TangentPoint(LineSegmentOf(P, F), G) = Q;VectorOf(P, Q) = 2*VectorOf(Q, F)", "coords": {} }, { "index": 269, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0269.png", "answer": "4", "fact_expr": "l: Line;C: Parabola;M: Point;F: Point;A: Point;Expression(C) = (x^2 = 4*y);Focus(C)=F;PointOnCurve(F,l);Slope(l)=sqrt(3);Intersection(l,C)={A,B};Intersection(TangentOnPoint(A,C),TangentOnPoint(B,C))=M;B:Point", "coords": {} }, { "index": 270, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0270.png", "answer": "sqrt(3)/4", "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;O: Origin;Expression(G) = (x^2/12 + y^2/3 = 1);OneOf(Focus(G)) = F1;PointOnCurve(P, G);MidPoint(LineSegmentOf(P, F1)) = M;PointOnCurve(M, yAxis);Distance(M, O) = Abs(LineSegmentOf(M, O))", "coords": {} }, { "index": 271, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0271.png", "answer": "{x=0,4*x-3*y-3=0}", "fact_expr": "l: Line;G: Parabola;H: Circle;Expression(G) = (x^2 = -4*y);Expression(H) = ((x - 1)^2 + (y - 2)^2 = 1);PointOnCurve(Focus(G),l);IsTangent(l,H)", "coords": {} }, { "index": 272, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 273, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 274, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/test/hyperbola/problem_0274.png", "answer": "2*sqrt(26)", "fact_expr": "C: Hyperbola;m: Number;P: Point;F1: Point;F2: Point;m>0;Expression(C) = (x^2/4 - y^2/m = 1);Expression(Asymptote(C)) = (y = pm*(sqrt(2)*x));LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F1)) = m - 1", "coords": {} }, { "index": 275, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0275.png", "answer": "(4*x^2 - y^2 - 6*y - 1 = 0)&Negation(y = 0)\ny = -1/4", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);P: Point;Q: Point;PointOnCurve(P, G) = True;PointOnCurve(Q, G) = True;Negation(Vertex(G) = P);Negation(Vertex(G) = Q);O: Origin;AngleOf(P, O, Q) = pi/4;l1: Line;l2: Line;PointOnCurve(P, l1) = True;PointOnCurve(Q, l2) = True;IsTangent(l1, G) = True;IsTangent(l2, G) = True;M: Point;Intersection(l1, l2) = M;A: Point;B: Point;Intersection(l1, xAxis) = A;Intersection(l2, xAxis) = B;l3: Line;PointOnCurve(Orthocenter(TriangleOf(A, B, M)), l3) = True;I: Point;Coordinate(I) = (0, -1/4);PointOnCurve(I, l3) = True", "coords": { "I": [ 0.0, -0.25 ] } }, { "index": 276, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0276.png", "answer": "3/4", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y = x^2/2);Intersection(l, G) = {A, B};A: Point;B: Point;IsPerpendicular(TangentOnPoint(A,G),TangentOnPoint(B,G)) = True;Coordinate(A) = (2, 2)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 277, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 278, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 279, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7841060161590576, "b": 0.3689577579498291, "orientation": "horizontal" }, "optimization": { "final_loss": 5.885034397579147e-07, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_0279.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-sqrt(3)/2, 0);Coordinate(F2) = (sqrt(3)/2, 0);Focus(G) = {F1, F2};Quadrant(P) = 1;PointOnCurve(P, G);Tan(AngleOf(P, F1, F2)) = 1/2;Tan(AngleOf(P, F2, F1)) = -2", "coords": { "F1": [ -0.8660254037844386, 0.0 ], "F2": [ 0.8660254037844386, 0.0 ] } }, { "index": 280, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0280.png", "answer": "[sqrt(3)/2,1)", "fact_expr": "C: Ellipse;b: Number;a: Number;O: Circle;A: Point;P: Point;B: Point;l1: Line;l2: Line;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);TangentOfPoint(P, O) = {l1, l2};TangentPoint(l1,O)=A;TangentPoint(l2,O)=B;AngleOf(A, P, B) = ApplyUnit(60, degree);PointOnCurve(P,C)", "coords": {} }, { "index": 281, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0281.png", "answer": "3*sqrt(3)", "fact_expr": "E: Ellipse;a: Number;F1: Point;M: Point;F2: Point;Expression(E) = (y^2/3 + x^2/a^2 = 1);RightFocus(E) = F2;LeftFocus(E)=F1;PointOnCurve(M, E);AngleOf(F1, M, F2) = (2*pi)/3", "coords": {} }, { "index": 282, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 283, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0283.png", "answer": "y=pm*2*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = (4/3)*x);RightFocus(G) = F;Intersection(H, G) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 284, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0284.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;In(A,C);F1: Circle;Center(F1) = F;Radius(F1) = LineSegmentOf(F,A);B: Point;D: Point;Intersection(F1, l) = {B, D};AngleOf(B, F, D) = ApplyUnit(90, degree);Area(TriangleOf(A, B, D)) = 4*sqrt(2)", "coords": {} }, { "index": 285, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0285.png", "answer": "6-sqrt(2)", "fact_expr": "G: Ellipse;A: Point;P: Point;F1: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(A) = (1, 1);LeftFocus(G) = F1;PointOnCurve(P, G)", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 286, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0286.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;A: Point;O: Origin;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);l1: Line;l2: Line;TangentOfPoint(OneOf(Endpoint(MajorAxis(C))),G) = {l1,l2};TangentPoint(l1,G)=A;TangentPoint(l2,G)=B;AngleOf(A, O, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 287, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0287.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Circle;PointOnCurve(Focus(G),H) = True;PointOnCurve(Vertex(G),H) = True;IsTangent(Directrix(G),H) = True", "coords": {} }, { "index": 288, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 289, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0289.png", "answer": "[sqrt(5)/2, sqrt(6)/2]", "fact_expr": "G: Curve;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;In(m, [-2, -1])", "coords": {} }, { "index": 290, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0290.png", "answer": "sqrt(2)/2", "fact_expr": "M: Point;Coordinate(M) = (1, 1);G: Line;Slope(G) = -1/2;PointOnCurve(M, G);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 291, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 292, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.33732376966822, "b": 1.7454107534785184, "orientation": "horizontal" }, "optimization": { "final_loss": 1.8920117601792354e-05, "converged": true, "iterations": 51 }, "output_path": "results/test/hyperbola/problem_0292.png", "answer": "x^2/16-y^2/9=1", "fact_expr": "G: Hyperbola;Coordinate(OneOf(Focus(G))) = (-5, 0);Expression(Asymptote(G))=(pm*4*y+3*x=0)", "coords": {} }, { "index": 293, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0293.png", "answer": "10", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 294, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0294.png", "answer": "(1, 3]", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G)) = True;Min(Abs(VectorOf(P, F2))^2/Abs(LineSegmentOf(P, F1))) = 8*a", "coords": {} }, { "index": 295, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0295.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/16 - y^2/b^2 = 1);b: Number;b>0;F: Point;LeftFocus(C) = F;M: Point;PointOnCurve(M, OneOf(Asymptote(C)));O: Origin;IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, F));Area(TriangleOf(O, M, F)) = 4", "coords": {} }, { "index": 296, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0296.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;O: Origin;PointOnCurve(O, G) = True;Slope(G) = 4/3;G: Line;Intersection(G, RightPart(C)) = A;A: Point;Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 297, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0297.png", "answer": "4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);G: Line;PointOnCurve(Focus(C), G);A: Point;B: Point;Intersection(G, C) = {A, B};L: Line;PointOnCurve(A, L);IsPerpendicular(L, Directrix(C));M: Point;FootPoint(L, Directrix(C)) = M;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, M))", "coords": {} }, { "index": 298, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0298.png", "answer": "1/2", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;F2: Point;D: Point;F1: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;UpperVertex(C)=A;MidPoint(LineSegmentOf(A, F2)) = D;IsPerpendicular(VectorOf(A,F2),VectorOf(D,F1))", "coords": {} }, { "index": 299, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 300, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0300.png", "answer": "(sqrt(3),2)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Point;PointOnCurve(G, C) ;Quadrant(G) = 1;H: Point;VectorOf(O, H) = VectorOf(O, G) + lambda*(VectorOf(G, F1)/Abs(VectorOf(G, F1)) + VectorOf(G, F2)/Abs(VectorOf(G, F2)));lambda: Number;Negation(lambda = 0);O: Origin;Intersection(LineOf(G,H), xAxis) = P;P : Point;Coordinate(P) = (sqrt(3)/3, 0)", "coords": { "P": [ 0.5773502691896257, 0.0 ] } }, { "index": 301, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0301.png", "answer": "2*sqrt(13)/13", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(3*x-2*y=0);Vertex(G)=Focus(H);Focus(G)=Vertex(H)", "coords": {} }, { "index": 302, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0302.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;D: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);In(A,Vertex(G));In(B,Vertex(G));L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,LineSegmentOf(A,B));IsPerpendicular(L2,LineSegmentOf(A,B));Intersection(L1, G) = C;Intersection(L2, G) = D;Negation(C=D);2*Abs(LineSegmentOf(B, D)) = 3*Abs(LineSegmentOf(A, C))", "coords": {} }, { "index": 303, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 304, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0304.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)", "coords": {} }, { "index": 305, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 4.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0305.png", "answer": "x^2/36-y^2/12=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 64);Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x+sqrt(3)*y=0)", "coords": {} }, { "index": 306, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0306.png", "answer": "{2/3,6}", "fact_expr": "W: Ellipse;m: Number;Expression(W) = (x^2/2 + y^2/m = 1);Eccentricity(W) = sqrt(6)/3", "coords": {} }, { "index": 307, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0307.png", "answer": "sqrt(13)/2", "fact_expr": "G: Hyperbola;Expression(G) = (9*x^2 - 4*y^2 = 36)", "coords": {} }, { "index": 308, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0308.png", "answer": "3/2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;M: Point;Expression(G)=(y=x^2/2);Intersection(l,G)={A,B};Abs(LineSegmentOf(A,B))=4;IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M", "coords": {} }, { "index": 309, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0309.png", "answer": "4", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 310, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 311, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 312, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0312.png", "answer": "x^2=pm*2*y", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;F: Point;Focus(C) = F;P: Point;Coordinate(P) = (x0, p/8);x0: Number;PointOnCurve(P,C) = True;Abs(LineSegmentOf(P,F)) = 5/8", "coords": {} }, { "index": 313, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0313.png", "answer": "6", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;PointOnCurve(RightFocus(G), H);IsPerpendicular(xAxis, H);Length(InterceptChord(H, G)) = m", "coords": {} }, { "index": 314, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0314.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "M: Parabola;Expression(M) = (y^2 = 3*x);l: Line;H: Point;Coordinate(H) = (3, 0);PointOnCurve(H, l);A: Point;B: Point;Intersection(l, M) = {A, B};O: Origin", "coords": { "H": [ 3.0, 0.0 ] } }, { "index": 315, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0315.png", "answer": "[3+2*sqrt(3),+oo)", "fact_expr": "G: Hyperbola;O: Origin;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);PointOnCurve(P, RightPart(G));Center(G)=O;LeftFocus(G)=F", "coords": {} }, { "index": 316, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 317, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0317.png", "answer": "{1,3}", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = p/2);PointOnCurve(A,G);PointOnCurve(B,G);Focus(G)=F;Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 4;Distance(MidPoint(LineSegmentOf(A,B)),H)=1", "coords": {} }, { "index": 318, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0318.png", "answer": "sqrt(15)/5", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;UpperVertex(G) = A;P: Point;Quadrant(P) = 1;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4*Abs(LineSegmentOf(F1, F2));Area(TriangleOf(P, F1, A))=2*Area(TriangleOf(P, F1, F2))", "coords": {} }, { "index": 319, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0319.png", "answer": "5", "fact_expr": "C: Parabola;p: Number;l: Line;A: Point;F: Point;M: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Expression(l)=(x=-5/4);PointOnCurve(M, C);Directrix(C)=l;PointOnCurve(A, l);IsPerpendicular(LineSegmentOf(M, A),l);Inclination(LineOf(A,F)) = pi/3", "coords": {} }, { "index": 320, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0320.png", "answer": "sqrt(37)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;Q: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {P, Q};IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(P, F1));Abs(LineSegmentOf(P, Q)) = (5/12)*Abs(LineSegmentOf(P, F1))", "coords": {} }, { "index": 321, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0321.png", "answer": "9", "fact_expr": "E: Parabola;l1:Line;l2:Line;A: Point;B: Point;C: Point;D: Point;F:Point;Expression(E) = (y^2 = x);Focus(E)=F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1,E)={A,B};Intersection(l2,E)={C,D}", "coords": {} }, { "index": 322, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0322.png", "answer": "2*x - y - 15 = 0", "fact_expr": "G: Hyperbola;H: LineSegment;P: Point;Expression(G) = (x^2/16 - y^2/4 = 1);Coordinate(P) = (8, 1);IsChordOf(H, G);MidPoint(H) = P", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 323, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0323.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G))=Focus(H);Distance(Focus(G),Asymptote(G))=1", "coords": {} }, { "index": 324, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_0324.png", "answer": "y=pm*(1/2)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(5)/2", "coords": {} }, { "index": 325, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0325.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;y0: Number;Coordinate(P) = (6, y0);PointOnCurve(P, G);F: Point;Focus(G) = F;Abs(LineSegmentOf(P, F)) = 8", "coords": {} }, { "index": 326, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 327, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0327.png", "answer": "6", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (y = x - 3);Focus(C) = F;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 328, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0328.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 2);p>0;Expression(H) = (y^2 = 2*(p*x));PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 329, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0329.png", "answer": "1/2", "fact_expr": "G: Hyperbola;H: Line;A: Point;a:Number;b:Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x);Coordinate(A) = (a, b);PointOnCurve(A,RightPart(G));Distance(A, H) = sqrt(2)", "coords": {} }, { "index": 330, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 331, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0331.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 332, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0332.png", "answer": "2", "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Distance(P,F1)=2;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)", "coords": {} }, { "index": 333, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 334, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 335, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0335.png", "answer": "15", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;M: Point;F1: Point;F2: Point;Coordinate(M) = (6, 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 336, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0336.png", "answer": "(-oo, -sqrt(6)/2]+[sqrt(6)/2, +oo)", "fact_expr": "C: Ellipse;l: Line;k: Number;Expression(C) = (x^2/2 + y^2 = 1);Expression(l) = (y = k*x + 2);IsIntersect(l, C)", "coords": {} }, { "index": 337, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0337.png", "answer": "2*sqrt(30)", "fact_expr": "C: Hyperbola;D:Hyperbola;b: Number;a: Number;G: Point;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Coordinate(G) = (2, 6);Expression(D)=(x^2/4-y^2/6=1);Asymptote(C)=Asymptote(D);PointOnCurve(G, C)", "coords": { "G": [ 2.0, 6.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 338, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0338.png", "answer": "2", "fact_expr": "O: Circle;Expression(O) = (x^2 + y^2 = 1);P: Point;Coordinate(P) = (2, 0);L1: Line;L2: Line;TangentOfPoint(P, O) = {L1, L2};TangentPoint(L1, O) = A;TangentPoint(L2, O) = B;A: Point;B: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z1: Line;Z2: Line;Asymptote(C) = {Z1, Z2};PointOnCurve(A, Z1);PointOnCurve(B, Z2)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 339, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0339.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Ellipse;a: Number;Expression(G) = (x^2/9 - y^2/3 = 1);a>0;Expression(H) = (y^2/4 + x^2/a^2 = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0340.png", "answer": "3", "fact_expr": "G: Hyperbola;a: Number;H: Circle;A: Point;M: Point;F2: Point;F1: Point;N: Point;a>0;Expression(G) = (-y^2/7 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;LeftVertex(G) = A;Center(H) = F2;Radius(H) = Abs(LineSegmentOf(F2, A));Intersection(H, RightPart(G)) = {M, N};PointOnCurve(N, PerpendicularBisector(LineSegmentOf(A, M)))", "coords": {} }, { "index": 341, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0341.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/5 + y^2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 342, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/test/hyperbola/problem_0342.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = sqrt(3)/2", "coords": {} }, { "index": 343, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 344, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0344.png", "answer": "25/8", "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 4);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}", "coords": { "A": [ 4.0, 4.0 ] } }, { "index": 345, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0345.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;XCoordinate(A) = 1;PointOnCurve(A, G);Distance(A, Focus(G)) = 5/2", "coords": {} }, { "index": 346, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0346.png", "answer": "sqrt(41)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);l: Line;Directrix(G) = l;C: Point;Coordinate(C) = (-3, -4);P: Point;PointOnCurve(P, G);d: Number;Distance(P, l) = d", "coords": { "C": [ -3.0, -4.0 ] } }, { "index": 347, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0347.png", "answer": "(5/2,+oo)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F2: Point;B: Point;P: Point;Q: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;LeftFocus(G)=F1;PointOnCurve(F1, H);IsPerpendicular(H,xAxis);Intersection(H, LeftPart(G)) = {A, B};Intersection(LineSegmentOf(A,F2),yAxis)=P;Intersection(LineSegmentOf(B, F2), yAxis) = Q;Perimeter(TriangleOf(P, Q, F2)) = 4", "coords": {} }, { "index": 348, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0348.png", "answer": "9*x-32*y+73=0", "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(M) = (-1, 2);IsChordOf(H, G);MidPoint(H) = M", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 349, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 350, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0350.png", "answer": "5", "fact_expr": "G: Hyperbola;H: Circle;U: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/15 = 1);Expression(H) = (y^2 + (x + 4)^2 = 4);Expression(U) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, U)", "coords": {} }, { "index": 351, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0351.png", "answer": "11", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;Coordinate(A) = (1, 3);P: Point;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 3.0 ] } }, { "index": 352, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0352.png", "answer": "(0,-1/4)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2)", "coords": {} }, { "index": 353, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0353.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;H: Ellipse;b: Number;Expression(G) = (x^2/2 - y^2/2 = 1);b>0;Expression(H) = (x^2/4 + y^2/b^2 = 1);PointOnCurve(Focus(H),Directrix(G))", "coords": {} }, { "index": 354, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/test/hyperbola/problem_0354.png", "answer": "-6", "fact_expr": "Eccentricity(C) = 3*sqrt(5)/5;C: Hyperbola;Expression(C) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;G: Parabola;m: Real;Expression(G) = (y^2 = 2*(m*x));LeftFocus(C) = Focus(G)", "coords": {} }, { "index": 355, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0355.png", "answer": "(0,3)", "fact_expr": "G: Parabola;F: Point;Expression(G) = (x^2 = 12*y);Focus(G)=F", "coords": {} }, { "index": 356, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0356.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;P: Point;y0: Number;Coordinate(P) = (6, y0);PointOnCurve(P, G);F: Point;Focus(G) = F;Abs(LineSegmentOf(P, F)) = 8", "coords": {} }, { "index": 357, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0357.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(VectorOf(P,F1))+Abs(VectorOf(P,F2))=4", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ] } }, { "index": 358, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0358.png", "answer": "7/2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 6*x);Coordinate(A) = (x1, y1);Coordinate(P) = (2, 2);Focus(G) = F;x1: Number;y1: Number;PointOnCurve(A, G)", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 359, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 360, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0360.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;P: Point;PointOnCurve(P,G) = True;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 361, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/test/hyperbola/problem_0361.png", "answer": "x^2/2-y^2/16=1", "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-3*sqrt(2), 0);LeftFocus(C) = F1;Eccentricity(C)=3", "coords": { "F1": [ -4.242640687119286, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 362, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0362.png", "answer": "Abs(a)/2", "fact_expr": "G: Parabola;a: Number;Negation(a=0);Expression(G) = (y^2 = a*x)", "coords": {} }, { "index": 363, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.818297058047652, "b": 1.8186428324319301, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 8.798975889096597e-06, "converged": true, "iterations": 98 }, "output_path": "results/test/ellipse/problem_0363.png", "answer": "{x^2/16+y^2/4=1,y^2/64+x^2/16=1}", "fact_expr": "O: Origin;Center(G) = O;G: Ellipse;PointOnCurve(Focus(G), axis);Length(MajorAxis(G)) = 2*Length(MinorAxis(G));P: Point;Coordinate(P) = (4, 0);PointOnCurve(P, G)", "coords": { "P": [ 4.0, 0.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 364, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0364.png", "answer": "(x^2-y^2/8=1)&(x>=1)", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x + 3)^2 = 4);A: Point;M: Circle;Coordinate(A) = (3, 0);PointOnCurve(A, M);IsOutTangent(C, M);M1: Point;Center(M) = M1", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 365, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0365.png", "answer": "2*sqrt(2)/3", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Slope(l) = k;k:Number;k>0;Intersection(l, C) = {A, B};LineSegmentOf(A, F)>LineSegmentOf(B, F);Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F))", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 366, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0366.png", "answer": "2", "fact_expr": "Q: Point;Coordinate(Q) = (2*sqrt(2), 0);G: Parabola;Expression(G) = (y = x^2/4);P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;PointOnCurve(P, G)", "coords": { "Q": [ 2.8284271247461903, 0.0 ] } }, { "index": 367, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 368, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 369, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0369.png", "answer": "25/4", "fact_expr": "F: Point;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;Focus(C) = F;A: Point;B: Point;PointOnCurve(A, C) = True;PointOnCurve(B, C) = True;VectorOf(A, F) = 4*VectorOf(F, B);Projection(A, Directrix(C)) = M;Projection(B, Directrix(C)) = N;M: Point;N: Point;Area(TriangleOf(M, F, N)) = 5", "coords": {} }, { "index": 370, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.33732376966822, "b": 1.7454107534785184, "orientation": "horizontal" }, "optimization": { "final_loss": 1.8920117601792354e-05, "converged": true, "iterations": 51 }, "output_path": "results/test/hyperbola/problem_0370.png", "answer": "{5/3,5/4}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/4)*x)", "coords": {} }, { "index": 371, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0371.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);PointOnCurve(Focus(G),H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 6;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 372, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0372.png", "answer": "", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);c: Number;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), xAxis) ;AngleOf(P, F2, F1) = ApplyUnit(45, degree)", "coords": {} }, { "index": 373, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.8993116218750659, "b": 1.324128928251209, "major_axis": "x", "x_coef": 3.999999999999999, "y_coef": 2.000000000000001 }, "optimization": { "final_loss": 0.013791354113027786, "converged": true, "iterations": 100 }, "output_path": "results/test/ellipse/problem_0373.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "G: Ellipse;H: Point;I: Point;Coordinate(H) = (sqrt(2), 1);Coordinate(I) = (sqrt(3), sqrt(2)/2);PointOnCurve(H, G);PointOnCurve(I, G)", "coords": { "H": [ 1.4142135623730951, 1.0 ], "I": [ 1.7320508075688772, 0.7071067811865476 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 374, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0374.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G)) = True;MidPoint(LineSegmentOf(P, F2)) = M;M: Point;O: Origin;Center(H) = O;Radius(H) = LineSegmentOf(O, F1);H: Circle;PointOnCurve(M, H) = True", "coords": {} }, { "index": 375, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0375.png", "answer": "5/3", "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Distance(F,Directrix(G)) = 2;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 376, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 377, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0377.png", "answer": "8/5", "fact_expr": "C: Parabola;p: Number;A: Point;P: Point;H: Point;0 < p;p < 4;Expression(C) = (x^2 = 2*(p*y));Coordinate(A) = (0, 4);Directrix(C) = l ;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, H),l);Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree);l:Line;FootPoint(LineSegmentOf(P, H),l)=H", "coords": { "A": [ 0.0, 4.0 ] } }, { "index": 378, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 379, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0379.png", "answer": "3", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;F1: Point;F2:Point;B: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(l)=sqrt(5)/2;LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,l);Intersection(l,LeftPart(G))=A;Intersection(l,RightPart(G))=B;PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(F1,B)))", "coords": {} }, { "index": 380, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 381, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0381.png", "answer": "sqrt(6)/3", "fact_expr": "H: Line;Expression(H) = (y = (sqrt(3)/3)*x);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};F: Point;RightFocus(G) = F;IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 382, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0382.png", "answer": "4*sqrt(2)", "fact_expr": "P: Point;Coordinate(P) = (2, 0);H: Line;PointOnCurve(P, H) = True;C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(H, C) = {A, B};A: Point;O: Origin;B: Point;H1: Line;Expression(H1) = (x=-2);Intersection(LineOf(O,A),H1) = M;Intersection(LineOf(O,B),H1) = N;M: Point;N: Point;G: Circle;IsDiameter(LineSegmentOf(M,N),G) = True", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 383, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0383.png", "answer": "y^2-4*x-y+4=0", "fact_expr": "M: Point;Coordinate(M) = (1, 1);H: Line;PointOnCurve(M, H);G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 384, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 385, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0385.png", "answer": "(sqrt(21)/7)+6", "fact_expr": "H: Ellipse;Expression(H) = (x^2/9 + y^2/8 = 1);F1: Point;F2: Point;LeftFocus(H) = F1;RightFocus(H) = F2;P: Point;PointOnCurve(P, H);Q: Point;lambda: Real;VectorOf(F1, P) = lambda*VectorOf(P, Q);Abs(VectorOf(P, Q)) = Abs(VectorOf(P, F2));G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);lambda>0", "coords": {} }, { "index": 386, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0386.png", "answer": "(1,sqrt(2))", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Line;PointOnCurve(RightFocus(G), H) = True;Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0387.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 388, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0388.png", "answer": "-2/3", "fact_expr": "M: Point;Coordinate(M) = (2, 1);PointOnCurve(M, l) ;l: Line;G: Ellipse;Expression(G) = (x^2/12 + y^2/4 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 389, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0389.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Point;Coordinate(H) = (3, -4);PointOnCurve(H, OneOf(Asymptote(G)))", "coords": { "H": [ 3.0, -4.0 ] } }, { "index": 390, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/test/parabola/problem_0390.png", "answer": "-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;Coordinate(M) = (1, 2);PointOnCurve(M, G);l: Line;A: Point;B: Point;Negation(A=B);Intersection(l, G) = {A, B};t: Number;Coordinate(Center(InscribedCircle(TriangleOf(M, A, B)))) = (1, t)", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 391, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0391.png", "answer": "4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C) = True;Abs(LineSegmentOf(P, F)) = 8;M: Point;MidPoint(LineSegmentOf(P, F)) = M", "coords": {} }, { "index": 392, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0392.png", "answer": "(sqrt(3)+sqrt(39))/6", "fact_expr": "l: Line;Gamma: Hyperbola;A: Point;O: Origin;B: Point;Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);a:Number;b:Number;a>0;b>0;Expression(l)=(x = sqrt(a^2 + b^2));Intersection(l, Gamma) = {A, B};DotProduct((VectorOf(A, B) + VectorOf(A, O)),VectorOf(B, O)) = 0", "coords": {} }, { "index": 393, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0393.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;Expression(G) = (2*x - y = 0);OneOf(Asymptote(C)) = G", "coords": {} }, { "index": 394, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 395, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0395.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;p: Number;F: Point;l: Line;H: Line;A: Point;B: Point;M: Point;N: Point;Q: Point;p > 0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);Intersection(H, C) = {A, B};IsPerpendicular(LineSegmentOf(A, M), l);IsPerpendicular(LineSegmentOf(B, N), l);FootPoint(LineSegmentOf(A, M), l) = M;FootPoint(LineSegmentOf(B, N), l) = N;MidPoint(LineSegmentOf(M, N)) = Q;Abs(LineSegmentOf(Q, F)) = 2", "coords": {} }, { "index": 396, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 397, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "up" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/test/parabola/problem_0397.png", "answer": "x-y-1=0", "fact_expr": "G: Parabola;p: Number;M: Point;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (2, 1);PointOnCurve(M, G);l:Line;IsTangent(l,G);TangentPoint(l,G)=M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 398, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 399, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0399.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));AngleOf(P, F1, F2) = ApplyUnit(45, degree)", "coords": {} }, { "index": 400, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 401, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0401.png", "answer": "(-12, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 + y^2/k = 1);k: Number;e: Number;Eccentricity(G) = e;In(e, (1, 2))", "coords": {} }, { "index": 402, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0402.png", "answer": "3", "fact_expr": "G: Parabola;H:Line;A: Point;C: Point;B: Point;D: Point;F:Point;l1:Line;l2:Line;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, H);Intersection(H,G)={A,B};PointOnCurve(A,l1);PointOnCurve(B,l2);IsPerpendicular(l1,xAxis);IsPerpendicular(l2,yAxis);FootPoint(l1,xAxis)=C;FootPoint(l2,yAxis)=D", "coords": {} }, { "index": 403, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0403.png", "answer": "{2*sqrt(41) - 2, 10}", "fact_expr": "G: Curve;a: Number;F1: Point;F2: Point;P: Point;Expression(G) = (y^2/16 + x^2/a = 1);FocalLength(G) = 10;PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 2", "coords": {} }, { "index": 404, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 405, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 406, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0406.png", "answer": "5", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(VectorOf(P, F1)) = 3;Abs(VectorOf(P, F2)) = 4", "coords": {} }, { "index": 407, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0407.png", "answer": "(1,0);sqrt(2)/2", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 = 4*x);F: Point;Focus(H) = F", "coords": {} }, { "index": 408, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0408.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/b^2 = 1);b: Number;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(F1, F2)) = 6", "coords": {} }, { "index": 409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0409.png", "answer": "7", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 6*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 410, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0410.png", "answer": "(pm*sqrt(5),0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 411, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 412, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0412.png", "answer": "2*sqrt(2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/2 = 1)", "coords": {} }, { "index": 413, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0413.png", "answer": "(0, 1/2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y)", "coords": {} }, { "index": 414, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0414.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 2);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 415, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 416, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0416.png", "answer": "-4", "fact_expr": "G: Parabola;P: Point;Q: Point;A: Point;L1:Line;L2:Line;Expression(G) = (x^2 = 2*y);PointOnCurve(P,G);PointOnCurve(Q,G);XCoordinate(P)=4;XCoordinate(Q)=-2;TangentOfPoint(P,G)=L1;TangentOfPoint(Q,G)=L2;Intersection(L1,L2)=A", "coords": {} }, { "index": 417, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0417.png", "answer": "[2,2*sqrt(2))", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(P) = (x0, y0);Focus(C)={F1,F2};x0:Number;y0:Number;0 b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*(c + x));LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*c;OneOf(Intersection(G, C)) = M;AngleOf(M, F1, F2) = 2*AngleOf(M, F2, F1)", "coords": {} }, { "index": 422, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0422.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;P: Point;A: Point;F: Point;Expression(C) = (x^2 = 4*y);Intersection(Directrix(C), axis) = A;PointOnCurve(P, C);Focus(C)=F", "coords": {} }, { "index": 423, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0423.png", "answer": "25/8", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G));l: Line;OneOf(Asymptote(G)) = l;Q: Point;Projection(P, l) = Q;Min(Abs(LineSegmentOf(P, F2)) + Abs(LineSegmentOf(P, Q))) = 5", "coords": {} }, { "index": 424, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0424.png", "answer": "[-sqrt(5), -1) + (-1, 1) + (1, sqrt(5)]", "fact_expr": "A: Point;Coordinate(A) = (0, 2);l: Line;PointOnCurve(A, l);G: Curve;Expression(G) = (x^2 - y^2 = 1);IsIntersect(l, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 425, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.8176298651812968, "b": 2.452642014841466, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4992967561765154e-06, "converged": true, "iterations": 283 }, "output_path": "results/test/hyperbola/problem_0425.png", "answer": "sqrt(10)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = 3*x)", "coords": {} }, { "index": 426, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0426.png", "answer": "2", "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(P, G);Center(G) = O;LeftFocus(G) = F", "coords": {} }, { "index": 427, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0427.png", "answer": "(sqrt(2)/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;Coordinate(A) = (a, 0);RightVertex(G) = A;P: Point;PointOnCurve(P, G) = True;O: Origin;AngleOf(A, P, O) = ApplyUnit(90, degree)", "coords": {} }, { "index": 428, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 429, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 430, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 431, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0431.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;a: Number;b: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);G: Line;O:Origin;PointOnCurve(O,G);P: Point;F: Point;Q: Point;a>0;b>0;Intersection(G, C) = {P, Q};OneOf(Focus(C)) = F;Tan(AngleOf(P, F, Q)) = 4/3;Abs(LineSegmentOf(Q, F)) = 5*Abs(LineSegmentOf(P, F));Negation(P=Q)", "coords": {} }, { "index": 432, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0432.png", "answer": "x^2/2 + y^2 = 1", "fact_expr": "C: Ellipse;b: Number;a: Number;O: Circle;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = 1);RightFocus(C) = F2;L:Line;TangentOnPoint(F2, O)=L;Length(InterceptChord(L,C))=sqrt(2)", "coords": {} }, { "index": 433, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0433.png", "answer": "1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Point;H: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (e, 1);Coordinate(H) = (-sqrt(3), sqrt(2));Eccentricity(C) = e;PointOnCurve(G, C);PointOnCurve(H, C)", "coords": { "H": [ -1.7320508075688772, 1.4142135623730951 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 434, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0434.png", "answer": "(x - 3)^2 + y^2 = 1", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/8 - y^2 = 1);Center(H) = RightFocus(G);IsTangent(H, Asymptote(G))", "coords": {} }, { "index": 435, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 436, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0436.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};O: Origin;G: Circle;Center(G) = O;Radius(G) = b;P: Point;Intersection(G, C) = P;Quadrant(P) = 1;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(P, F2))", "coords": {} }, { "index": 437, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0437.png", "answer": "10", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a = 1);Expression(H) = (-6*x + x^2 + y^2 + 8 = 0);OneOf(Focus(G)) = Center(H);Length(MinorAxis(G))=8", "coords": {} }, { "index": 438, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0438.png", "answer": "(2/3)*sqrt(3)", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;p>0;Expression(G) = (x^2/3 - 16*y^2/p^2 = 1);Expression(H) = (y^2 = 2*(p*x));PointOnCurve(LeftFocus(G), Directrix(H))", "coords": {} }, { "index": 439, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0439.png", "answer": "8", "fact_expr": "E: Ellipse;Expression(E) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(E) = {F1, F2};l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, E) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 440, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0440.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 3);IsTangent(H, Asymptote(G))", "coords": {} }, { "index": 441, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.23606797749979, "major_axis": "x", "x_coef": 8.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0441.png", "answer": "x^2/3-y^2/5=1", "fact_expr": "H: Ellipse;Expression(H) = (x^2/8 + y^2/5 = 1);Focus(H) = Vertex(C);C: Hyperbola;Focus(C) = Vertex(H)", "coords": {} }, { "index": 442, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0442.png", "answer": "2*x-y-15=0", "fact_expr": "P: Point;Coordinate(P) = (8, 1);G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);H: LineSegment;IsChordOf(H,G) = True;MidPoint(H) = P", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 443, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0443.png", "answer": "5/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;G: Circle;Center(G) = F;B: Point;Quadrant(B) = 1;TangentPoint(G, OneOf(Asymptote(C))) = B;Slope(LineOf(A, B)) = 1/2", "coords": {} }, { "index": 444, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.496487617492676, "direction": "right" }, "optimization": { "final_loss": 5.969127983007638e-07, "converged": true, "iterations": 103 }, "output_path": "results/test/parabola/problem_0444.png", "answer": "13/2", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(P) = (2, 6);PointOnCurve(P, C);Focus(C) = F", "coords": { "P": [ 2.0, 6.0 ] } }, { "index": 445, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0445.png", "answer": "3*sqrt(10)", "fact_expr": "G: Hyperbola;H: Line;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/9 = 1);Expression(H) = (x = 1);LeftFocus(G)=F1;RightFocus(G)=F2;Intersection(H,OneOf(Asymptote(G)))=P", "coords": {} }, { "index": 446, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0446.png", "answer": "4", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = 2*x);Coordinate(A) = (x1, y1) ;Coordinate(B) = (x2, y2);Focus(G) = F;PointOnCurve(F, l);x1 + x2 = 3;Intersection(l, G) = {A, B};Negation(A = B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 447, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0447.png", "answer": "(0,1)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4)", "coords": {} }, { "index": 448, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 12.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0448.png", "answer": "{30,10}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/144 = 1);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (13, 0);Distance(P, A) = 20;B: Point;Coordinate(B) = (-13, 0)", "coords": { "A": [ 13.0, 0.0 ], "B": [ -13.0, 0.0 ] } }, { "index": 449, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_0449.png", "answer": "x^2/4-y^/12=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 450, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0450.png", "answer": "x+2y-3=0", "fact_expr": "C: Ellipse;Expression(C) = (x^2 + 2*y^2 = 4);P: Point;Coordinate(P) = (1, 1);L: Line;PointOnCurve(P, L) = True;Intersection(L, C) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 451, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0451.png", "answer": "1/16", "fact_expr": "H: Circle;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);G: Parabola;Expression(G) = (y = a*x^2);a: Number;a>0;IsTangent(H, Directrix(G))", "coords": {} }, { "index": 452, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0452.png", "answer": "sqrt(2) - 1", "fact_expr": "E: Parabola;p: Number;C: Ellipse;a: Number;b: Number;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(E) = F;RightFocus(C) = F;A: Point;B: Point;Intersection(C, E) = {A, B};PointOnCurve(F, LineOf(A, B))", "coords": {} }, { "index": 453, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 454, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0454.png", "answer": "", "fact_expr": "G: Hyperbola;H: Curve;a: Number;Expression(G) = (x^2/9 - y^2/4 = 1);Expression(H) = (y = a*x^2 + 1/3);L1:Line;L2:Line;Asymptote(G)={L1,L2};IsTangent(L1,H);IsTangent(L2,H)", "coords": {} }, { "index": 455, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0455.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);Slope(H) = -1;A: Point;B: Point;Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 456, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0456.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x0, -2);x0:Number;PointOnCurve(A,G);Distance(A,Focus(G))=3*Distance(A,yAxis)", "coords": {} }, { "index": 457, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 458, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0458.png", "answer": "6", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/12 = 1);Focus(C) = {F1,F2};PointOnCurve(P, C);Abs(LineSegmentOf(P, F2)) = 3", "coords": {} }, { "index": 459, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0459.png", "answer": "4/3", "fact_expr": "l: Line;C: Parabola;F: Point;M: Point;P: Point;Q: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {P, Q};Intersection(l, Directrix(C)) = M;VectorOf(F, M) = 3*VectorOf(F, P)", "coords": {} }, { "index": 460, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 461, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0461.png", "answer": "x^2/4+y^2=1", "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(I) = (1, sqrt(3)/2);PointOnCurve(I, H);Focus(G)=Focus(H)", "coords": { "I": [ 1.0, 0.8660254037844386 ] } }, { "index": 462, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0462.png", "answer": "63/4", "fact_expr": "P: Point;Coordinate(P) = (5, 0);C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);M: Point;N: Point;PointOnCurve(M,RightPart(C)) ;PointOnCurve(N,RightPart(C)) ;IsPerpendicular(VectorOf(P, M), VectorOf(P, N))", "coords": { "P": [ 5.0, 0.0 ] } }, { "index": 463, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0463.png", "answer": "4*sqrt(6)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Abs(LineSegmentOf(P, F2)) = 5", "coords": {} }, { "index": 464, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0464.png", "answer": "2", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;M: Point;A: Point;B: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = M;RightFocus(G) = F;PointOnCurve(F, l);IsPerpendicular(xAxis, l);Intersection(l, G) = {A,B};IsPerpendicular(LineSegmentOf(M, A), LineSegmentOf(M, B))", "coords": {} }, { "index": 465, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0465.png", "answer": "x^2-y^2/3=1", "fact_expr": "G:Hyperbola;a:Number;b:Number;a>0;b>0;C:Parabola;e:Number;Expression(G)=(x^2/a^2-y^2/b^2=1);Expression(C)=(y^2=8*x);Eccentricity(G)=e;e=2;OneOf(Focus(G))=Focus(C)", "coords": {} }, { "index": 466, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0466.png", "answer": "2", "fact_expr": "C1: Parabola;C2: Circle;M: Point;A: Point;B: Point;Coordinate(M) = (1, 1);Expression(C1) = (y^2 = 8*x);Expression(C2) = (x^2+y^2-4*x+3=0);PointOnCurve(A, C1);PointOnCurve(B, C2)", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 467, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 468, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0468.png", "answer": "(1/2, 2/3)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;k: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;PointOnCurve(A, G);Slope(G) = k;Intersection(G, C) = {A, B};Projection(B, xAxis) = F;RightFocus(C) = F;1/3 < k;k < 1/2", "coords": {} }, { "index": 469, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 470, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0470.png", "answer": "(-1,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x)", "coords": {} }, { "index": 471, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0471.png", "answer": "", "fact_expr": "P: Point;Coordinate(P) = (2, 1);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 472, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0472.png", "answer": "{0,-1,-4/5}", "fact_expr": "G: Line;Expression(G) = (y = x*(a + 1) - 1);H: Curve;Expression(H) = (y^2 = a*x);a: Real;NumIntersection(G, H)=1", "coords": {} }, { "index": 473, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0473.png", "answer": "14", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b = 1);Expression(Asymptote(G)) = (y = pm*x)", "coords": {} }, { "index": 474, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0474.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 475, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 476, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0476.png", "answer": "17/8", "fact_expr": "G: Parabola;A: Point;Expression(G) = (y = 2*x^2);Coordinate(A) = (1, 2);PointOnCurve(A, G);Focus(G) = F;F: Point", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 477, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0477.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;HalfFocalLength(G) = c;c: Number;PointOnCurve(RightFocus(G), L);Slope(L) = 1;L: Line;NumIntersection(L, RightPart(G)) = 2;H: Parabola;Expression(H) = (y^2 = 4*c*x);Length(InterceptChord(Directrix(H), G)) = (2*sqrt(2)/3)*b*e^2;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 478, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "y", "x_coef": 27.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0478.png", "answer": "y^2/4-x^2/5=1", "fact_expr": "C: Hyperbola;F:Ellipse;P:Point;Expression(F) = (x^2/27 + y^2/36 = 1);Coordinate(P) = (sqrt(15), 4);Focus(C) = Focus(F);PointOnCurve(P,C)", "coords": { "P": [ 3.872983346207417, 4.0 ] } }, { "index": 479, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0479.png", "answer": "4*x+9*y-13=0", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;M: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(M) = (1, 1);Intersection(G, H) = {A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A, B)) = M;PointOnCurve(M,H)", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 480, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0480.png", "answer": "2*sqrt(3)/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F,l);Slope(l)=a/b;L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(L1,l)=A;Intersection(L2,l)=B;VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 481, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0481.png", "answer": "5*sqrt(5)/3", "fact_expr": "L: Line;Slope(L) = 2;G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);RightFocus(G) = F2;F2: Point;PointOnCurve(F2, L) = True;Intersection(L, G) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B), G) = True", "coords": {} }, { "index": 482, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0482.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 483, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0483.png", "answer": "-1", "fact_expr": "C: Parabola;p: Number;Z: Line;A: Point;F: Point;M: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(M) = (-1, 4);Focus(C) = F;PointOnCurve(M, Z);IsPerpendicular(Z, yAxis);Intersection(Z, C) = A;Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(A, M));Intersection(LineOf(A, F), C) = {A, B}", "coords": { "M": [ -1.0, 4.0 ] } }, { "index": 484, "success": false, "error": "Circle from diameter: cannot parse diameter pattern" }, { "index": 485, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0485.png", "answer": "20", "fact_expr": "C: Hyperbola;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 - y^2/4 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, LeftPart(C));Abs(LineSegmentOf(A, F1)) = 12", "coords": {} }, { "index": 486, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 487, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0487.png", "answer": "8*sqrt(6)/9", "fact_expr": "l: Line;G: Parabola;p: Number;F: Point;A: Point;B: Point;m:Number;p>0;Expression(G) = (y^2 = 2*p*x);Expression(l)=(y=-x+m);m>=0;m<1;Coordinate(F) = (-1, 0);Focus(G) = F;Intersection(l,G) = {A, B};Negation(A = B)", "coords": { "F": [ -1.0, 0.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 488, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0488.png", "answer": "(0,1)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)", "coords": {} }, { "index": 489, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 490, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0490.png", "answer": "7/2", "fact_expr": "G: Ellipse;H: Line;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);IsPerpendicular(H, xAxis);IsIntersect(H, G);OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 491, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0491.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Line;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsPerpendicular(xAxis, H);Intersection(H, G) = P", "coords": {} }, { "index": 492, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0492.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightFocus(G), OneOf(Asymptote(G))) = 2*a", "coords": {} }, { "index": 493, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0493.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)", "coords": {} }, { "index": 494, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0494.png", "answer": "5/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A,G) = True;PointOnCurve(B,G) = True;VectorOf(A,F)= 3*VectorOf(F,B)", "coords": {} }, { "index": 495, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0495.png", "answer": "4", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M,G);LeftFocus(G)=F1;Distance(M,F1)=2;MidPoint(LineSegmentOf(M,F1))=N", "coords": {} }, { "index": 496, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 497, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0497.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(OneOf(Asymptote(G)))=sqrt(3)", "coords": {} }, { "index": 498, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 499, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0499.png", "answer": "x^2/4 + y^2 = 1", "fact_expr": "C: Circle;N: Point;P: Point;M: Point;O: Origin;l: Line;Expression(C) = (x^2 + y^2 = 4);PointOnCurve(M, C);PointOnCurve(M, l);IsPerpendicular(l, xAxis);FootPoint(l, xAxis) = N;VectorOf(N, P) = VectorOf(P, M)", "coords": {} }, { "index": 500, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0500.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;c: Number;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(C) = c;Eccentricity(C) = e", "coords": {} }, { "index": 501, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0501.png", "answer": "(2, 1)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (2, 3);WhenMin(LineSegmentOf(P, A) + LineSegmentOf(P, F))", "coords": { "A": [ 2.0, 3.0 ] } }, { "index": 502, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0502.png", "answer": "33", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1);M: Point;PointOnCurve(M, G) = True;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;Abs(LineSegmentOf(M, F1)) = 17", "coords": {} }, { "index": 503, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 504, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0504.png", "answer": "7/3", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/7 = 1);PointOnCurve(P, C);Focus(C) = {F1, F2};Radius(InscribedCircle(TriangleOf(P,F1,F2)))=1;Quadrant(P)=1", "coords": {} }, { "index": 505, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0505.png", "answer": "18", "fact_expr": "C: Parabola;l1: Line;l2: Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (x^2 = 2*y);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1, C) = {A, B};Intersection(l2, C) = {D,E}", "coords": {} }, { "index": 506, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0506.png", "answer": "16", "fact_expr": "G: Ellipse;H: Circle;P: Point;M: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = ((x + 1)^2 + (y - 3)^2 = 1);LeftFocus(G)=F1;PointOnCurve(P,G);PointOnCurve(M, H)", "coords": {} }, { "index": 507, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/test/hyperbola/problem_0507.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = pi/3;Eccentricity(G) = sqrt(6)/2", "coords": {} }, { "index": 508, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0508.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;C: Point;Intersection(Directrix(G), xAxis) = C;H: Circle;Center(H) = F;Radius(H) = p;A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 509, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0509.png", "answer": "(0,sqrt(2)-1]", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F2: Point;F1: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),E);PointOnCurve(F1,LineSegmentOf(A,B));Abs(VectorOf(F2,A)-VectorOf(F2,B)) >= Abs(VectorOf(F2,A)+VectorOf(F2,B))", "coords": {} }, { "index": 510, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/test/parabola/problem_0510.png", "answer": "pi/3", "fact_expr": "P: Point;Coordinate(P) = (1, 2);E: Parabola;Expression(E) = (y^2 = 2*p*x);p: Number;p>0;PointOnCurve(P, E);M: Point;Coordinate(M) = (1, 0);PointOnCurve(M, l);Slope(l) > 0;l: Line;Intersection(l, E) = {A, B};A: Point;B: Point;VectorOf(A, M) = 3*VectorOf(M, B)", "coords": { "P": [ 1.0, 2.0 ], "M": [ 1.0, 0.0 ] } }, { "index": 511, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0511.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;UpperVertex(G)=B;RightFocus(G) = F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F));a > b;b > 0", "coords": {} }, { "index": 512, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0512.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;O: Origin;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 3", "coords": {} }, { "index": 513, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 514, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0514.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;F: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;UpperVertex(G) = B;Length(MinorAxis(G)) = 4;Eccentricity(G)=sqrt(5)/5", "coords": {} }, { "index": 515, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0515.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(A, G);Focus(G) = F;Abs(LineSegmentOf(A,F))=4;AngleOf(O, F, A) = ApplyUnit(120, degree)", "coords": {} }, { "index": 516, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0516.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;H: Parabola;p: Number;M: Point;F: Point;m:Number;m>0;Expression(G) = (x^2/2 - y^2/b^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);Coordinate(M) = (1, m);PointOnCurve(M, H);Focus(H) = F;Distance(M, F) = 5;Distance(F,OneOf(Asymptote(G))) = 2*sqrt(2)", "coords": {} }, { "index": 517, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 518, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0518.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Distance(P, L1)*Distance(P, L2) = b^2/3", "coords": {} }, { "index": 519, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0519.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;a: Number;b: Number;A1: Circle;M: Point;A: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;Center(A1)=A;Length(Radius(A1))=b;Intersection(A1, OneOf(Asymptote(C))) = {M, N};AngleOf(M, A, N) = ApplyUnit(60, degree)", "coords": {} }, { "index": 520, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 5.656854249492381, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0520.png", "answer": "2", "fact_expr": "G: Hyperbola;A: Point;x0: Number;y0:Number;Expression(G) = (x^2/4 - y^2/32 = 1);Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));Distance(A,RightFocus(G)) = 2*x0", "coords": {} }, { "index": 521, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0521.png", "answer": "{0,1}", "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Number;G: Parabola;Expression(G) = (y^2 = 4*x);NumIntersection(H, G) = 1", "coords": {} }, { "index": 522, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0522.png", "answer": "3*x-y-11=0", "fact_expr": "G: Parabola;H: LineSegment;P: Point;Expression(G) = (y^2 = 6*x);Coordinate(P) = (4, 1);PointOnCurve(P,H);MidPoint(H)=P;IsChordOf(H,G)", "coords": { "P": [ 4.0, 1.0 ] } }, { "index": 523, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0523.png", "answer": "2/3", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};VectorOf(A, F) = VectorOf(F, B)/2", "coords": {} }, { "index": 524, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0524.png", "answer": "9/8", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 8*x);Length(LineSegmentOf(A, B)) = 6;MidPoint(LineSegmentOf(A, B)) = M;IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 525, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0525.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F2: Point;P: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(O,F2),G);Intersection(G, OneOf(Asymptote(C))) = {O, P};AngleOf(O, P, F1) = ApplyUnit(45, degree)", "coords": {} }, { "index": 526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0526.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;O: Origin;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(Focus(C), G);Slope(G) = 1;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 527, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0527.png", "answer": "sqrt(14)/4", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;I: Line;B: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(I) = (y = b/3);Intersection(I, G) = {B, C};IsDiameter(LineSegmentOf(B,C),H);RightFocus(G)=F;PointOnCurve(F,H);F:Point", "coords": {} }, { "index": 528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0528.png", "answer": "1/3", "fact_expr": "l: Line;Gamma: Parabola;O: Origin;F: Point;A: Point;B: Point;Expression(Gamma) = (y^2 = 4*x);Expression(l)=(sqrt(3)*x - y - sqrt(3) = 0);Intersection(l,Gamma) = {A, B};Intersection(l,xAxis)=F;LineSegmentOf(O,F)=lambda*LineSegmentOf(O,A)+mu*LineSegmentOf(O,B);lambda<=mu;lambda:Number;mu:Number", "coords": {} }, { "index": 529, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0529.png", "answer": "9/10", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Expression(Asymptote(G)) = (3*x + pm*(4*y) = 0)", "coords": {} }, { "index": 530, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0530.png", "answer": "4", "fact_expr": "l1: Line;Expression(l1) = (4*x - 3*y + 16 = 0);l2: Line;Expression(l2) = (x = -1);G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);d1: Number;Distance(P, l1) = d1;d2: Number;Distance(P, l2) = d2", "coords": {} }, { "index": 531, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0531.png", "answer": "3", "fact_expr": "G: Curve;H: Curve;Expression(G) = (y^2 = 2*x);Expression(H) = (y^2 + (x - 2)^2 = 4)", "coords": {} }, { "index": 532, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0532.png", "answer": "(3, 0)\ny^2 = 12*x", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);C: Parabola;F: Point;RightFocus(G) = F;O: Origin;Vertex(C) = O;Focus(C) = F", "coords": {} }, { "index": 533, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 534, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 535, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 536, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_0536.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 537, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0537.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;Gamma: Circle;P: Point;F1: Point;Q: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C)=F1;PointOnCurve(P, C);IsDiameter(LineSegmentOf(P,F1),Gamma);PointOnCurve(Q, Gamma);O:Origin", "coords": {} }, { "index": 538, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 539, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0539.png", "answer": "x^2/3-y^2=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (y^2 + (x - 2)^2 = 1);Center(E)=RightFocus(C);IsTangent(E, Asymptote(C))", "coords": {} }, { "index": 540, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 541, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0541.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(2)/2))", "coords": {} }, { "index": 542, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0542.png", "answer": "[sqrt(5),sqrt(6)]", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;O: Origin;M: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 1 - x);Eccentricity(G)=e;Intersection(H,G)={M,N};Range(e)=[1/sqrt(3),1/sqrt(2)];IsPerpendicular(LineSegmentOf(O,M),LineSegmentOf(O,N));e:Number", "coords": {} }, { "index": 543, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 544, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 545, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0545.png", "answer": "12", "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;b>0;Expression(G) = (x^2/9 - y^2/b^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 546, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0546.png", "answer": "12", "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, 3);Distance(H, Focus(G)) = 5", "coords": { "H": [ 2.0, 3.0 ] } }, { "index": 547, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0547.png", "answer": "5", "fact_expr": "G:Hyperbola;a: Number;P: Point;F2: Point;F1: Point;Expression(G) = (-y^2/9 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));Expression(OneOf(Asymptote(G))) = (3*x - y = 0);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F2)) = 3", "coords": {} }, { "index": 548, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0548.png", "answer": "5/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;PointOnCurve(F1, LineSegmentOf(A, B)) = True;Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi;Coordinate(A) = (x1, y1);x1: Number;y1: Number;Coordinate(B) = (x2, y2);x2: Number;y2: Number", "coords": {} }, { "index": 549, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 550, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0550.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Expression(H) = (x^2 + y^2 = a^2);L: Line;TangentOfPoint(F2,H) = L;Intersection(L,LeftPart(G)) = M;M: Point;AngleOf(F1, M, F2) = ApplyUnit(45, degree)", "coords": {} }, { "index": 551, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0551.png", "answer": "[3, 27/8]", "fact_expr": "G: Ellipse;b: Number;H: Line;M: Point;N: Point;F2: Point;F1: Point;lambda: Number;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {M, N};Min(Length(LineSegmentOf(M, N))) = 3;Abs(VectorOf(N, F2)) = lambda*Abs(VectorOf(F2, M));In(lambda, [1, 2]);IsChordOf(LineSegmentOf(M, N), G)", "coords": {} }, { "index": 552, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0552.png", "answer": "3*sqrt(2)/2", "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 553, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0553.png", "answer": "1/9", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*p*x);M: Point;a: Real;m:Number;m>0;p: Number;p>0;Distance(M,Focus(H))=5;G: Hyperbola;Expression(G) = (-y^2 + x^2/a=1);A:Point;LeftVertex(G)=A;Coordinate(M) = (1, m);PointOnCurve(M, H);IsParallel(OneOf(Asymptote(G)),LineOf(A,M))", "coords": {} }, { "index": 554, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0554.png", "answer": "3/2", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/3 + y^2/2 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = sqrt(3)/2", "coords": {} }, { "index": 555, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 556, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 4.0, 1.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0556.png", "answer": "x^2/12-y^2/4=1", "fact_expr": "c: Hyperbola;Expression(Asymptote(c)) = (x + pm*sqrt(3)*y = 0);G: Circle;Expression(G) = (-2*y - 8*x + x^2 + y^2 + 16 = 0);PointOnCurve(RightFocus(c), G)", "coords": {} }, { "index": 557, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0557.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "G: Ellipse;O: Origin;Center(G) = O;Eccentricity(G) = 1/2;H: Parabola;Expression(H) = (y^2 = -4*x);l: Line;Directrix(H) = l;PointOnCurve(OneOf(Focus(G)), l)", "coords": {} }, { "index": 558, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0558.png", "answer": "1/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F: Point;RightFocus(G) = F;l: Line;PointOnCurve(F, l);Q: Point;Intersection(l, MinorAxis(G)) = Q;M: Point;N: Point;Intersection(l, G) = {M, N};lambda: Number;mu: Number;VectorOf(F, M) = lambda*VectorOf(M, Q);VectorOf(F, N) = mu*VectorOf(N, Q)", "coords": {} }, { "index": 559, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/test/hyperbola/problem_0559.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x))", "coords": {} }, { "index": 560, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0560.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, RightPart(C));O: Origin;Abs(LineSegmentOf(P, F)) = Abs(LineSegmentOf(O, F));Cos(AngleOf(O, F, P)) = 1/4", "coords": {} }, { "index": 561, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0561.png", "answer": "6", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G) = True;H1: Circle;Expression(H1) = (y^2 + (x + 1)^2 = 1);H2: Circle;Expression(H2) = (y^2 + (x - 1)^2 = 1);Q: Point;R: Point;PointOnCurve(Q, H1) = True;PointOnCurve(R, H2) = True", "coords": {} }, { "index": 562, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 563, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0563.png", "answer": "(pm*2*sqrt(2), 0)\ny = pm*(sqrt(15)/3)*x", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/8 + y^2/5 = 1);Eccentricity(G) = 2*sqrt(6)/3;Vertex(G) = Focus(H)", "coords": {} }, { "index": 564, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0564.png", "answer": "[sqrt(6/3),1)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftVertex(C) = A1;RightVertex(C) = A2;A1: Point;A2: Point;PointOnCurve(P, C) = True;P: Point;AngleOf(A1, P, A2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 565, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142164151010692, "b": 1.4142164151010692, "orientation": "horizontal" }, "optimization": { "final_loss": 2.2160979279094996e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_0565.png", "answer": "(-4,0)+(0,4)", "fact_expr": "G:Hyperbola;F1: Point;F2: Point;m:Number;P:Point;Coordinate(F1)=(-2,0);Coordinate(F2)=(2, 0);Distance(P,F1)-Distance(P,F2)=m;Locus(P)=G", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ] } }, { "index": 566, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0566.png", "answer": "(0, 1)", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2", "coords": {} }, { "index": 567, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 568, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0568.png", "answer": "(16/5,-9/5)", "fact_expr": "l: Line;G: Ellipse;P: Point;A: Point;B: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Expression(l)=(x-y+1=0);Intersection(l,G)={A,B};PointOnCurve(P,G);WhenMax(Area(TriangleOf(P,A,B)))", "coords": {} }, { "index": 569, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.660254037844387, "b": 5.0, "major_axis": "y", "x_coef": 25.0, "y_coef": 75.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0569.png", "answer": "25*sqrt(3)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/75 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 570, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0570.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);H: Line;PointOnCurve(RightFocus(G), H);IsPerpendicular(H, xAxis);A: Point;B: Point;l1: Line;l2: Line;Asymptote(G) = {l1, l2};Intersection(H, l1) = A;Intersection(H, l2) = B", "coords": {} }, { "index": 571, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0571.png", "answer": "7", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y = x^2/4);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);y1 + y2 = 5", "coords": {} }, { "index": 572, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 573, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0573.png", "answer": "5", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, C) = True;P: Point;PointOnCurve(P, OverlappingLine(LineSegmentOf(F1, A))) = True;DotProduct(VectorOf(A, F2) + VectorOf(A, P), VectorOf(F2, P)) = 0", "coords": {} }, { "index": 574, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0574.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B:Point;P: Point;F:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);l:Line;l1:Line;l2:Line;RightFocus(C)=F;PointOnCurve(F, l);IsPerpendicular(l,l1);FootPoint(l,l1)=B;Intersection(l,yAxis)=P;Intersection(l,l2)=A;Between(P,A,B)=True;Abs(LineSegmentOf(O, A)) = (5/3)*a;OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1=l2)", "coords": {} }, { "index": 575, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0575.png", "answer": "2/3", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A:Point;B: Point;F: Point;e:Number;k:Number;a > b;b > 0;k>0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;Slope(G)=k;LeftVertex(C)=A;Intersection(G,C)={A,B};Projection(B, xAxis) = F;Eccentricity(C) = e ;e= 1/3", "coords": {} }, { "index": 576, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0576.png", "answer": "(sqrt(2)-1)/4", "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;A: Point;P: Point;O: Origin;c:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;Expression(l)=(x=2*c);PointOnCurve(P,l);AngleOf(A,P,O)=ApplyUnit(45,degree)", "coords": {} }, { "index": 577, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0577.png", "answer": "sqrt(30)/5", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(C) = A;A1: Circle;Center(A1) = A;Radius(A1) = b;M: Point;N: Point;Intersection(A1, OneOf(Asymptote(C))) = {M, N};O: Origin;VectorOf(O, M) = (3/2)*VectorOf(O, N)", "coords": {} }, { "index": 578, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0578.png", "answer": "sqrt(21)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;L: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, L);IsPerpendicular(L, OneOf(Asymptote(G)));FootPoint(L, OneOf(Asymptote(G))) = P;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 579, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 580, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0580.png", "answer": "ApplyUnit(60,degree)", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 581, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0581.png", "answer": "5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;P1: Point;P2: Point;PointOnCurve(P1, C) ;PointOnCurve(P2, C) ;YCoordinate(P1) = y1;YCoordinate(P2) = y2;y1^2 + y2^2 = 8;y1:Number;y2:Number", "coords": {} }, { "index": 582, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0582.png", "answer": "10*sqrt(5)", "fact_expr": "G: Parabola;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, B));Orthocenter(TriangleOf(A,O,B))=Focus(G)", "coords": {} }, { "index": 583, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0833333358168602, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0583.png", "answer": "y=-1/12", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/3)", "coords": {} }, { "index": 584, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0584.png", "answer": "sqrt(5)", "fact_expr": "C1: Parabola;Expression(C1) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C1) = F;C2: Hyperbola;Expression(C2) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;OneOf(Intersection(C1, OneOf(Asymptote(C2)))) = A;A: Point;IsPerpendicular(LineSegmentOf(A, F), xAxis) = True", "coords": {} }, { "index": 585, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0585.png", "answer": "9*sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 586, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0586.png", "answer": "y=-1/16", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2)", "coords": {} }, { "index": 587, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0587.png", "answer": "(x+1)^2+y^2=7", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);H: Circle;Focus(G) = Center(H);Radius(H) = sqrt(7)", "coords": {} }, { "index": 588, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0588.png", "answer": "(2 - sqrt(2))/2", "fact_expr": "l: Line;G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (x + y + sqrt(2) = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 589, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0589.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/2 = 1)", "coords": {} }, { "index": 590, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0590.png", "answer": "3*x+4*y-7=0", "fact_expr": "G: Ellipse;H: LineSegment;P:Point;Expression(G) = (x^2/4 + y^2/3=1);MidPoint(H)=P;Coordinate(P) = (1,1);IsChordOf(H,G)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 591, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0591.png", "answer": "sqrt(13)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;P: Point;H: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;l:Line;PointOnCurve(F, l);IsPerpendicular(l, OneOf(Asymptote(C)));FootPoint(l, OneOf(Asymptote(C)))=H;PointOnCurve(P, C);VectorOf(F, P) = 3*VectorOf(F, H)", "coords": {} }, { "index": 592, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0592.png", "answer": "y=pm*(sqrt(3)*x)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(G) = 2", "coords": {} }, { "index": 593, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0593.png", "answer": "5*x^2/4-5*y^2/16=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x - 4);RightFocus(G)=F;PointOnCurve(F, H);NumIntersection(H,G)=1", "coords": {} }, { "index": 594, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 595, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0595.png", "answer": "(4,pm*4)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);D: Point;PointOnCurve(D, G) = True;Distance(D, Focus(G)) = 5", "coords": {} }, { "index": 596, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0596.png", "answer": "8", "fact_expr": "l: Line;E: Parabola;C: Point;D: Point;A: Point;B: Point;Expression(E) = (y^2 = 4*x);PointOnCurve(Focus(E), l);Intersection(l, E) = {A, B};Intersection(TangentOnPoint(A,E), yAxis) = C;Intersection(TangentOnPoint(B,E),yAxis)=D", "coords": {} }, { "index": 597, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0597.png", "answer": "3*sqrt(2)-5", "fact_expr": "C: Ellipse;E: Circle;M: Point;N: Point;F1:Point;F2:Point;LeftFocus(C)=F1;RightFocus(C)=F2;Expression(C) = (x^2/4+y^2/3=1);Expression(E)=((x-4)^2+(y-3)^2=1);PointOnCurve(M,C);PointOnCurve(N,E)", "coords": {} }, { "index": 598, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0598.png", "answer": "3*sqrt(2)/2", "fact_expr": "G: Parabola;H: Line;P0: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y + 4 = 0);PointOnCurve(P0, G)", "coords": {} }, { "index": 599, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0599.png", "answer": "sqrt(3) - 1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;O: Origin;l: Line;PointOnCurve(O, l);Intersection(l, C) = {A, B};A: Point;B: Point;2*Abs(LineSegmentOf(F, O)) = Abs(LineSegmentOf(A, B));AngleOf(B, A, F) = pi/6", "coords": {} }, { "index": 600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0600.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(Focus(C), l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 4;Distance(MidPoint(LineSegmentOf(A, B)), yAxis) = 1", "coords": {} }, { "index": 601, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 602, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0602.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 603, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 604, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 605, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0605.png", "answer": "y^2=20*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*m*x);m: Number;m>0;G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);Distance(Focus(H),OneOf(Asymptote(G))) = 3", "coords": {} }, { "index": 606, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0606.png", "answer": "y=-4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 16*y)", "coords": {} }, { "index": 607, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0607.png", "answer": "5/4", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;M: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(C)) = 8;RightFocus(C)=F;PointOnCurve(M,OneOf(Asymptote(C)));IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, F));Area(TriangleOf(O,M,F)) = 6", "coords": {} }, { "index": 608, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 609, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0609.png", "answer": "28/3", "fact_expr": "l: Line;G: Ellipse;B: Point;F2: Point;A: Point;F1: Point;Expression(G) = (x^2/9 + y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 610, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0610.png", "answer": "5", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;B: Point;F: Point;O: Origin;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F));Abs(LineSegmentOf(O, F)) = 5;Abs(LineSegmentOf(B, F)) = 8", "coords": {} }, { "index": 611, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0611.png", "answer": "{sqrt(2),1}", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = k*x + 1);NumIntersection(H, G) = 1", "coords": {} }, { "index": 612, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0612.png", "answer": "x^2/4+y^2=1", "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;F: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0,-2);Eccentricity(E) = sqrt(3)/2;Focus(E) = F;Slope(LineOf(A,F))=2*sqrt(3)/3", "coords": { "A": [ 0.0, -2.0 ] } }, { "index": 613, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 614, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0614.png", "answer": "4", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;RightVertex(C) = A;PointOnCurve(P, C);Max(Abs(LineSegmentOf(P, F))) = 3;Min(Abs(LineSegmentOf(P, F))) = 1", "coords": {} }, { "index": 615, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0615.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (-2, 3);Distance(H, Focus(G)) = 5", "coords": { "H": [ -2.0, 3.0 ] } }, { "index": 616, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0616.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);F1: Point;F2: Point;LeftFocus(G)=F1;RightFocus(G)=F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 617, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 618, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0618.png", "answer": "x^2/4 - y^2/12=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)", "coords": {} }, { "index": 619, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0619.png", "answer": "", "fact_expr": "C: Parabola;p: Number;M: Point;N: Point;A: Point;F: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(A) = (0, 3);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(M,l);IsPerpendicular(LineSegmentOf(M,N),l);Min(Abs(LineSegmentOf(M, A)) + Abs(LineSegmentOf(M, N))) = 2;FootPoint(LineSegmentOf(M,N),l)=N", "coords": { "A": [ 0.0, 3.0 ] } }, { "index": 620, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 621, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0621.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);Slope(l) = 1;Intersection(l, G) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 622, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 623, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0623.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;H: Line;F: Point;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = pi/4;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 624, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0624.png", "answer": "[2, +oo)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);NumIntersection(H, RightPart(G)) = 1", "coords": {} }, { "index": 625, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0625.png", "answer": "{y^2 = 8*x,y^2 = -16*x}", "fact_expr": "G: Parabola;m: Number;H: Line;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G), H) = 3", "coords": {} }, { "index": 626, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 627, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0627.png", "answer": "x=2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = m*x);m: Number;E: Ellipse;Expression(E) = (x^2/9 + y^2/5 = 1);Focus(C) = LeftFocus(E)", "coords": {} }, { "index": 628, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 629, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 630, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0630.png", "answer": "12", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/4 - y^2/m = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 631, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0631.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;K: Point;Intersection(Directrix(C), xAxis) = K;A: Point;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 632, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0632.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);Slope(H) = 1;A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 633, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0633.png", "answer": "3", "fact_expr": "C: Parabola;l: Line;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;PointOnCurve(F, l);Inclination(l)=ApplyUnit(60,degree);Intersection(l,C) = {A, B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 634, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0634.png", "answer": "4/3", "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/5 - y^2/4 = 1);Focus(G) = {F1,F2};PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2))", "coords": {} }, { "index": 635, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 636, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 637, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0637.png", "answer": "8 - 4*sqrt(2)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 16;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 638, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0638.png", "answer": "y=pm*(3/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 639, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0639.png", "answer": "12", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Center(G), LineSegmentOf(A, B));F1: Point;OneOf(Focus(G) )= F1", "coords": {} }, { "index": 640, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0640.png", "answer": "y=(15/8)*x+1", "fact_expr": "C: Parabola;a: Number;P: Point;F: Point;a>0;Expression(C) = (x^2 = a*y);Coordinate(P) = (2*a, 4*a);PointOnCurve(P, C);Focus(C) = F;Distance(P, F) = 17", "coords": {} }, { "index": 641, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 642, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7071067811865476, "b": 0.5773502691896257, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0642.png", "answer": "y=pm*(sqrt(6)/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (2*x^2 - 3*y^2 = 1)", "coords": {} }, { "index": 643, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0643.png", "answer": "3*sqrt(5)", "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;l: Line;d: Number;Expression(C) = (y^2 = 16*x);Coordinate(A) = (1, 6);Focus(C) = F;PointOnCurve(P, C);Directrix(C) = l;Distance(P, l) = d", "coords": { "A": [ 1.0, 6.0 ] } }, { "index": 644, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 645, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0645.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = sqrt(3)/3", "coords": {} }, { "index": 646, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0646.png", "answer": "2*sqrt(7)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;G: Circle;Diameter(G) = Length(RealAxis(C));l1: Line;l2: Line;Intersection(G,l1) = P;P: Point;Quadrant(P) = 2;IsParallel(LineSegmentOf(P,A) , l2) = True;O: Origin;Area(TriangleOf(O,P,A)) = sqrt(3);OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1=l2)", "coords": {} }, { "index": 647, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 648, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0648.png", "answer": "3", "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;Focus(G) = F;PointOnCurve(F, H) = True;H: Line;OneOf(Intersection(H, G)) = A;A: Point;Intersection(H, Directrix(G)) = B;B: Point;VectorOf(B, A) = 2*VectorOf(A, F);Abs(VectorOf(B, A)) = 4", "coords": {} }, { "index": 649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_0649.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 650, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0650.png", "answer": "-3/4", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;P: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 1);PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=P;IsChordOf(LineSegmentOf(A,B),C)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 651, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0651.png", "answer": "(0, 4)", "fact_expr": "H: Parabola;Expression(H) = (x^2 = 2*(p*y));p: Number;p>0;G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1);PointOnCurve(OneOf(Vertex(G)), Directrix(H)) = True", "coords": {} }, { "index": 652, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0652.png", "answer": "3", "fact_expr": "G: Parabola;p: Number;A: Point;N: Point;M: Point;F: Point;B: Point;l: Line;l1: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = ((7/2)*p, 0);Focus(G) = F;Directrix(G) = l;PointOnCurve(M, l1);IsPerpendicular(l, l1);FootPoint(l, l1) = N;Intersection(LineSegmentOf(A, N), LineSegmentOf(M, F)) = B;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(M, F));Area(TriangleOf(A, B, M)) = (9*sqrt(2))/2;PointOnCurve(M, G)", "coords": {} }, { "index": 653, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0653.png", "answer": "y = -1/4", "fact_expr": "G: Parabola;Expression(G) = (y = x^2)", "coords": {} }, { "index": 654, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 655, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 656, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.1622776601683795, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0656.png", "answer": "y^2=-16*x", "fact_expr": "C: Hyperbola;G: Parabola;O: Origin;Expression(C) = (x^2/10 - y^2/6 = 1);Vertex(G) = O;SymmetryAxis(G)=xAxis;Focus(G)=LeftFocus(C)", "coords": {} }, { "index": 657, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0657.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Line;M: Point;F2: Point;N: Point;F1: Point;Expression(G) = (x^2/4 - y^2/3 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, H);Intersection(H, LeftPart(G)) = {M, N}", "coords": {} }, { "index": 658, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0658.png", "answer": "1", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 659, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0659.png", "answer": "2*sqrt(5)/5", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;P: Point;Q: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;PointOnCurve(O, l);Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, Q)) = a;IsPerpendicular(LineSegmentOf(A, P), LineSegmentOf(P, Q))", "coords": {} }, { "index": 660, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0660.png", "answer": "2", "fact_expr": "G: Ellipse;H: Circle;l: Line;m: Number;n: Number;J: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(H) = (x^2 + y^2 = 4);Expression(l) = (m*x + n*y = 4);Coordinate(J) = (m, n);NumIntersection(l, H)=0;PointOnCurve(J, L);L:Line", "coords": {} }, { "index": 661, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0661.png", "answer": "11*sqrt(5)/5", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x + 2*y - 12 = 0);PointOnCurve(P, G);d1: Number;d2: Number;Distance(P, Directrix(G)) = d1;Distance(P, H) = d2", "coords": {} }, { "index": 662, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0662.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 3*y);PointOnCurve(A,G);YCoordinate(A)=5/4", "coords": {} }, { "index": 663, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 664, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 665, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 666, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0666.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;D: Point;F: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;Intersection(Directrix(C), yAxis) = D;PointOnCurve(F, G);Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(A, D));Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(A, F)) + 4", "coords": {} }, { "index": 667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0667.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F2: Point;A: Point;F1: Point;c:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y =-(b/a)*x);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, RightPart(G));Abs(LineSegmentOf(A,F1))=(11/2)*a-c;HalfFocalLength(G)=c;IsParallel(LineOf(A,F2),H)", "coords": {} }, { "index": 668, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0668.png", "answer": "pm*2*sqrt(2)", "fact_expr": "l: Line;C: Parabola;G: Line;A: Point;E: Point;B: Point;H: Point;F: Point;k: Number;Expression(C) = (y^2 = 8*x);Expression(G) = (x = -2);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,G);IsPerpendicular(L2,G);FootPoint(L1,G)=E;FootPoint(L2,G)=H;Abs(LineSegmentOf(A, E)) = 2*Abs(LineSegmentOf(B, H));Slope(l) = k", "coords": {} }, { "index": 669, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 670, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0670.png", "answer": "(1,sqrt(2)]", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c", "coords": {} }, { "index": 671, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0671.png", "answer": "32/3", "fact_expr": "C: Hyperbola;G: Line;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 - y^2/16 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(Focus(C),G);IsParallel(G,Asymptote(C));Intersection(G,C)=M", "coords": {} }, { "index": 672, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0672.png", "answer": "1/2", "fact_expr": "C: Ellipse;a: Number;G: Point;Expression(C) = (y^2/3 + x^2/a^2 = 1);Coordinate(G) = (1, 0);OneOf(Focus(C)) = G", "coords": { "G": [ 1.0, 0.0 ] } }, { "index": 673, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, -3.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0673.png", "answer": "{y^2=9*x, x^2=(-1/3)*y}", "fact_expr": "G: Parabola;SymmetryAxis(G) = axis;O: Origin;Vertex(G) = O;H: Circle;Expression(H) = (6*y - 2*x + x^2 + y^2 + 9 = 0);PointOnCurve(Center(H), G)", "coords": {} }, { "index": 674, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0674.png", "answer": "1+(3*sqrt(2))/4", "fact_expr": "C: Ellipse;G: Line;A: Point;F: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};k: Number;Abs(LineSegmentOf(A,F))+2*Abs(LineSegmentOf(B,F)) >= k", "coords": {} }, { "index": 675, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 676, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0676.png", "answer": "-4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p > 0;PointOnCurve(Focus(G), l) = True;l: Line;Intersection(l, G) = {A, B};A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number", "coords": {} }, { "index": 677, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 678, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0678.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1,F2};c: Number;FocalLength(G) = 2*c;P: Point;PointOnCurve(P, G) = True;M: Point;H: Circle;Expression(H) = (x^2 + y^2 = a^2);TangentPoint(LineOf(P,F1),H) = M;MidPoint(LineSegmentOf(P,F1)) = M", "coords": {} }, { "index": 679, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 680, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_0680.png", "answer": "{9/4,4}", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (y^2/3 + x^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 681, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 682, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 683, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0683.png", "answer": "sqrt(3)", "fact_expr": "C: Ellipse;E: Circle;N: Point;M: Point;Expression(C) = (x^2/9 + y^2/8 = 1);Expression(E) = (y^2 + (x - 1)^2 = 1);PointOnCurve(M, C);PointOnCurve(N, E);IsTangent(LineOf(M,N),E)", "coords": {} }, { "index": 684, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0684.png", "answer": "[1, +oo)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;Coordinate(Focus(G)) = (1, 0);H: Point;PointOnCurve(H, G)", "coords": {} }, { "index": 685, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0685.png", "answer": "y=pm*2*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;I: Line;k: Number;A: Point;B: Point;F:Point;a>0;b>0;Negation(k=0);Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(I) = (y = k*x);Intersection(I, G) = {A, B};IsDiameter(LineSegmentOf(A,B),H);PointOnCurve(F,H);RightFocus(G)=F;Area(TriangleOf(A,B,F))=4*a^2", "coords": {} }, { "index": 686, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0686.png", "answer": "4*sqrt(3)", "fact_expr": "E: Ellipse;Expression(E) = (x^2/3 + y^2 = 1);F1: Point;Focus(E) = {F1, F2};l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, E) = {A, B};F2: Point", "coords": {} }, { "index": 687, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0687.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;k: Number;O: Origin;M: Point;F1:Point;F2: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);k>0;Expression(H) = (y = k*x);LeftFocus(G) =F1;RightFocus(G)=F2;Intersection(H, G) = {M, N};Abs(LineSegmentOf(O, M)) = Abs(LineSegmentOf(O, F2));Area(TriangleOf(M,F2,N))=(Abs(LineSegmentOf(M,F2))+Abs(LineSegmentOf(N,F2)))^2/16", "coords": {} }, { "index": 688, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0688.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (9*p/2, 0);Abs(LineSegmentOf(Q, F)) = 2*Abs(LineSegmentOf(P, F));Area(TriangleOf(P, Q, F)) = 8*sqrt(3)", "coords": {} }, { "index": 689, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0689.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(F, P) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 690, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 691, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0691.png", "answer": "(-2, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)", "coords": {} }, { "index": 692, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0692.png", "answer": "(3/5,-2/5)", "fact_expr": "H: Line;Expression(H) = (x - y = 1);G: Ellipse;Expression(G) = (x^2/3 + y^2/2 = 1);Intersection(H,G) = {A,B};A: Point;B: Point", "coords": {} }, { "index": 693, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 694, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0694.png", "answer": "4", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (3, y1);y1:Number;PointOnCurve(P,G)", "coords": {} }, { "index": 695, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0695.png", "answer": "25", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G) = True;Focus(G) = {F1,F2};F1: Point;F2: Point;Distance(P,F1) * Distance(P,F2) = m;m:Number", "coords": {} }, { "index": 696, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0696.png", "answer": "sqrt(5)+4", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;F: Point;M:Number;N:Number;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (-1, 1);Coordinate(B) = (3, 2);PointOnCurve(P, G);Focus(G) = F;Min(Distance(P,A)+Distance(P,H))=M;Min(Abs(LineSegmentOf(P, B)) + Abs(LineSegmentOf(P, F))) = N", "coords": { "A": [ -1.0, 1.0 ], "B": [ 3.0, 2.0 ] } }, { "index": 697, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0697.png", "answer": "12+6*sqrt(3)", "fact_expr": "C: Ellipse;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/36 + y^2/9 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(A, C)", "coords": {} }, { "index": 698, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0698.png", "answer": "{(y^2=3*x), (y^2=-3*x)}", "fact_expr": "G: Parabola;O: Origin;H: Circle;Vertex(G) = O;SymmetryAxis(G) = xAxis;Expression(H) = (x^2 + y^2 = 4);Length(InterceptChord(G, H)) = 2*sqrt(3)", "coords": {} }, { "index": 699, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0699.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;H: Circle;a: Number;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 = 0);RightFocus(G) = Center(H)", "coords": {} }, { "index": 700, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0700.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (-a, 0);Coordinate(B) = (0, b);LeftFocus(G) = F;In(A, Vertex(G));In(B, Vertex(G));Distance(F, LineOf(A, B)) = b/sqrt(7)", "coords": {} }, { "index": 701, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 702, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 703, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 704, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0704.png", "answer": "2*sqrt(2)", "fact_expr": "H: Parabola;Expression(H) = (x^2 = 2*p*y);p: Number;p>0;G: Hyperbola;Expression(G) = (-x^2 + y^2 = 1);PointOnCurve(OneOf(Focus(G)), Directrix(H)) = True", "coords": {} }, { "index": 705, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0705.png", "answer": "64*sqrt(3)/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/100 + y^2/64 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 706, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0706.png", "answer": "sqrt(3)/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;P: Point;Coordinate(P) = (3/2, y0);y0: Number;Focus(G) = F;F: Point;Distance(P, F) = 2;O: Origin;PointOnCurve(P, G) = True", "coords": {} }, { "index": 707, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0707.png", "answer": "(3,2*sqrt(3))", "fact_expr": "m: Line;G: Parabola;n: Line;P: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Quadrant(P) = 1;TangentOnPoint(P, G) = m;PointOnCurve(P,n);IsParallel(xAxis, n);C:Line;PointOnCurve(F,C);IsParallel(C,m);Intersection(C,n)=M;Abs(LineSegmentOf(P,M))=4", "coords": {} }, { "index": 708, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 709, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0709.png", "answer": "8*sqrt(3)/3", "fact_expr": "C: Ellipse;F1: Point;P: Point;F2: Point;Expression(C) = (x^2/16+y^2/8=1);Focus(C)={F1,F2};PointOnCurve(P, C);AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 710, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0710.png", "answer": "-1/3", "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);Expression(H) = (x^2/4 + y^2 = 1);LeftFocus(H) = F1;RightFocus(H) = F2;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 711, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0711.png", "answer": "(0,2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y)", "coords": {} }, { "index": 712, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0712.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;a: Number;b: Number;c:Number;G: Circle;H: Line;F1: Point;O: Origin;N: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(C)=F1;RightFocus(C) = F2;PointOnCurve(F1, H);TangentPoint(H, G) = M;Intersection(H,RightPart(C))=N;Abs(LineSegmentOf(O, N)) = c;F2:Point;c>0", "coords": {} }, { "index": 713, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0713.png", "answer": "18", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F,H);Intersection(H,G)={A,B}", "coords": {} }, { "index": 714, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0714.png", "answer": "4/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Cos(AngleOf(F1, P, F2)) = 1/2", "coords": {} }, { "index": 715, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0715.png", "answer": "[sqrt(3)/3,1)", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;I: Point;O: Origin;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Incenter(TriangleOf(P, F1, F2)) = I", "coords": {} }, { "index": 716, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0716.png", "answer": "{x^2/10-y^2/6=1, x^2/10-y^2/6=-1}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/3 = 1);Z: Hyperbola;Asymptote(G) = Asymptote(Z);FocalLength(Z) = 8", "coords": {} }, { "index": 717, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0717.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l1: Line;l2: Line;Asymptote(G) = {l1, l2};P: Point;Quadrant(P) = 1;PointOnCurve(P, l1);IsPerpendicular(l2, LineSegmentOf(P, F1));IsParallel(l2, LineSegmentOf(P, F2))", "coords": {} }, { "index": 718, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0718.png", "answer": "[sqrt(5),oo)", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Slope(l)=2;PointOnCurve(F,l);NumIntersection(l, RightPart(G)) = 1", "coords": {} }, { "index": 719, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0719.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = x^2);Expression(H) = (x + y + 2 = 0);PointOnCurve(P, G);WhenMin(Distance(P, H))", "coords": {} }, { "index": 720, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 721, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0721.png", "answer": "[sqrt(2),sqrt(21)/3)", "fact_expr": "G: Hyperbola;H: Ellipse;D:Ellipse;a: Number;a<=1;Expression(H) = (x^2/4 + y^2/16 = 1);Expression(D) = (y^2/4 + (a*x^2)/16 = 1);PointOnCurve(Focus(G),xAxis);PointOnCurve(Intersection(D,H),Asymptote(G))", "coords": {} }, { "index": 722, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0722.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2 + y^2/5 = 1);Vertex(G) = Center(H);Focus(G) = RightVertex(H)", "coords": {} }, { "index": 723, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0723.png", "answer": "2/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;x1: Number;x2: Number;y1: Number;y2: Number;PointOnCurve(F1,L) = True;Intersection(L,G) = {A,B};Coordinate(A) = (x1,y1);Coordinate(B) = (x2,y2);Radius(InscribedCircle(TriangleOf(A,B,F2))) = 1;Abs(LineSegmentOf(F1,F2)) = 2;Abs(y1-y2) = 3;A: Point;B: Point", "coords": {} }, { "index": 724, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0724.png", "answer": "x+4*y-5=0", "fact_expr": "G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(M) = (1, 1);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 725, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0725.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Negation(A=B);M: Point;Intersection(Directrix(C), xAxis) = M;Tan(AngleOf(A, M, B)) = 2*sqrt(6)", "coords": {} }, { "index": 726, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0726.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 6", "coords": {} }, { "index": 727, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0727.png", "answer": "5", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P, yAxis) = 4", "coords": {} }, { "index": 728, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0728.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/5 + x^2/a^2 = 1);a: Number;H: Point;Coordinate(H) = (3, 0);RightFocus(G) = H", "coords": { "H": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 729, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0729.png", "answer": "8", "fact_expr": "G: Parabola;H: Curve;k: Number;M: Point;F: Point;Expression(G) = (y^2 = 8*x);k>0;Expression(H) = (x*y = k);Intersection(G, H) = M;Focus(G) = F;Distance(M, F) = 4", "coords": {} }, { "index": 730, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0730.png", "answer": "y=pm*3*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/36 = 1)", "coords": {} }, { "index": 731, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 732, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 733, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 734, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0734.png", "answer": "[sqrt(3)/3,sqrt(2)/2]", "fact_expr": "M: Ellipse;b: Number;a: Number;c:Number;P: Point;F1: Point;F2: Point;e: Number;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M)=F1;RightFocus(M)=F2;PointOnCurve(P,M);Range(Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2)))=[2*c^2,3*c^2];c=sqrt(a^2-b^2);Eccentricity(M)=e", "coords": {} }, { "index": 735, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 736, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/test/hyperbola/problem_0736.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = sqrt(3);OneOf(Directrix(G))=Directrix(H)", "coords": {} }, { "index": 737, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0737.png", "answer": "(1,sqrt(2))", "fact_expr": "E: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P,Asymptote(E));Abs(Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2))) = 2*b", "coords": {} }, { "index": 738, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0738.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Circle;Expression(H) = (m + 6*x + x^2 + y^2 = 0);m: Real;IsTangent(Directrix(G), H)", "coords": {} }, { "index": 739, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0739.png", "answer": "8", "fact_expr": "M: Point;Coordinate(M) = (sqrt(3), 0);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);H: Line;Expression(H) = (y = k*(x + sqrt(3)));A: Point;B: Point;Intersection(G,H) = {A,B};k:Number", "coords": { "M": [ 1.7320508075688772, 0.0 ] } }, { "index": 740, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0740.png", "answer": "7", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2);Distance(P, F1) = 3", "coords": {} }, { "index": 741, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 0.9927864074707031, "direction": "right" }, "optimization": { "final_loss": 0.00015326803259085864, "converged": true, "iterations": 44 }, "output_path": "results/test/parabola/problem_0741.png", "answer": "5", "fact_expr": "G: Parabola;p: Number;P: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (4, 4);PointOnCurve(P, G);Focus(G) = F", "coords": { "P": [ 4.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 742, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 743, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0743.png", "answer": "9/4", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 2*(VectorOf(F, B));IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 744, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0744.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/7 = 1);Focus(G) = Endpoint(MajorAxis(H));PointOnCurve(Focus(H),Directrix(G))", "coords": {} }, { "index": 745, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0746.png", "answer": "y = pm*(2/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)", "coords": {} }, { "index": 747, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0747.png", "answer": "4*sqrt(3)/3", "fact_expr": "r1: Hyperbola;r2:Hyperbola;O: Origin;A: Point;B: Point;x1:Number;x2:Number;y1:Number;y2:Number;a:Number;b:Number;a>0 ;b>0;Expression(r1) = (x^2/a^2-y^2/b^2=1);Expression(r2) = (y^2/b^2-x^2/a^2=1);PointOnCurve(A,r1);PointOnCurve(B,r2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1^2 - x2^2 = a^2;Slope(LineOf(O, A))*Slope(LineOf(O,B)) = 1/3", "coords": {} }, { "index": 748, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_0748.png", "answer": "2*x+pm*y=0", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 749, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0749.png", "answer": "(x^2+(y-1)^2=3/4)&(x>0,y>0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);A: Point;UpperVertex(G) = A;F: Point;LeftFocus(G) = F;C: Point;MidPoint(LineSegmentOf(O, A)) = C;O: Origin;P: Point;Quadrant(P) = 1;Abs(LineSegmentOf(C, P)) = (sqrt(3)/2)*Abs(LineSegmentOf(O, C))", "coords": {} }, { "index": 750, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0750.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A1: Point;A2: Point;LeftVertex(G) = A1;RightVertex(G) = A2;P: Point;PointOnCurve(P, LeftPart(G));Slope(LineOf(P, A1)) = -1;Slope(LineOf(P, A2)) = -1/3", "coords": {} }, { "index": 751, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0751.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(G))=2;Length(ImageinaryAxis(G)) = 4", "coords": {} }, { "index": 752, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0752.png", "answer": "y = pm*2*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;P: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;OneOf(Endpoint(ImageinaryAxis(C))) = B;PointOnCurve(P, LeftPart(C));Min(Perimeter(TriangleOf(B, P, F))) = 4*Length(RealAxis(C))", "coords": {} }, { "index": 753, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0753.png", "answer": "20", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Slope(H)=2;PointOnCurve(Focus(G),H);Intersection(H,G)={A,B}", "coords": {} }, { "index": 754, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0754.png", "answer": "2", "fact_expr": "M: Point;PointOnCurve(M, G) = True;G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;MidPoint(LineSegmentOf(M,F)) = P;Distance(P, yAxis) = 2", "coords": {} }, { "index": 755, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0755.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1, y1);Coordinate(A) = (2, 2);Focus(G) = F;x1:Number;y1:Number;PointOnCurve(P, G)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 756, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -4.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0756.png", "answer": "x=13/8", "fact_expr": "O1: Circle;Expression(O1) = (x^2 + y^2 = 1);O2: Circle;Expression(O2) = (y^2 + (x - 4)^2 = 4);P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;TangentOfPoint(P, O1) = LineOf(P, A);TangentOfPoint(P, O2) = LineOf(P, B);TangentPoint(LineOf(P, A), O1) = A;TangentPoint(LineOf(P, B), O2) = B;A: Point;B: Point;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 757, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0757.png", "answer": "2/7", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;m: Number;n: Number;Coordinate(P) = (m, n);PointOnCurve(P, G);Distance(P, Focus(G)) = 8*m", "coords": {} }, { "index": 758, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0758.png", "answer": "4*sqrt(5)/3", "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 759, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0759.png", "answer": "(0, sqrt(3))", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;e: Number;Eccentricity(G) = e;In(e, (1, 2))", "coords": {} }, { "index": 760, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0760.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;A: Point;Q: Point;H:Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;Center(H)=A;Radius(H)=sqrt(2)*a;Intersection(H,RightPart(C))={P,Q};AngleOf(P, A, Q) = pi/2", "coords": {} }, { "index": 761, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0761.png", "answer": "8", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;M: Point;B: Point;Expression(C) = (y^2 = 6*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(Directrix(C), xAxis) = M;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(M, F))", "coords": {} }, { "index": 762, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 763, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0763.png", "answer": "(2,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 764, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0764.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(G)=Focus(H)", "coords": {} }, { "index": 765, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0765.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;B: Point;A: Point;O: Origin;Expression(G) = (x^2 - y^2/2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0", "coords": {} }, { "index": 766, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_0766.png", "answer": "y^2/4-x^2=1", "fact_expr": "E: Hyperbola;Expression(Asymptote(E)) = (y = pm*2*x);Coordinate(Focus(E)) = (0, pm*sqrt(5))", "coords": {} }, { "index": 767, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0767.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 2)", "coords": {} }, { "index": 768, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0768.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, xAxis) = True;G: Line;Expression(G) = (y = m + 2*x);m: Number;Negation(m = 0);NumIntersection(G, C) = 1;PointOnCurve(P, G) = True", "coords": {} }, { "index": 769, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0769.png", "answer": "6", "fact_expr": "C: Parabola;G: Line;P: Point;B: Point;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (1, 0);Focus(C) = F;PointOnCurve(P, G);Intersection(G, C) = {A, B};2*VectorOf(B, P) = VectorOf(P, A)", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 770, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0770.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 771, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0771.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;l:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1,F2};PointOnCurve(F2,l);IsPerpendicular(l,xAxis);Intersection(l,G)={A,B};Radius(InscribedCircle(TriangleOf(A,B,F1)))=a", "coords": {} }, { "index": 772, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 773, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0773.png", "answer": "10", "fact_expr": "F: Point;LeftFocus(E) = F;E: Ellipse;Expression(E) = (x^2/16 + y^2/12 = 1);P: Point;PointOnCurve(P, E) = True;A: Point;Coordinate(A) = (1, sqrt(3))", "coords": { "A": [ 1.0, 1.7320508075688772 ] } }, { "index": 774, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0774.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 775, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.2628633975982666, "b": 1.8162860737044515, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.6997004191253788e-06, "converged": true, "iterations": 101 }, "output_path": "results/test/ellipse/problem_0775.png", "answer": "", "fact_expr": "C: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);A: Point;Coordinate(A)=(1, 2);PointOnCurve(A, C)", "coords": { "A": [ 1.0, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 776, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 777, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0777.png", "answer": "9", "fact_expr": "G: Ellipse;m: Number;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/m = 1);Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Focus(G) = {F1,F2}", "coords": { "F1": [ -4.0, 0.0 ], "F2": [ 4.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 778, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0778.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(F2, LineSegmentOf(A, B))", "coords": {} }, { "index": 779, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0779.png", "answer": "16/3", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y = x^2/4);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 780, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 781, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 782, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 783, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0783.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 6;Abs(LineSegmentOf(B, F)) = 3", "coords": {} }, { "index": 784, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 785, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0785.png", "answer": "5/2", "fact_expr": "G: Ellipse;a: Number;B: Point;P: Point;e:Number;a>1;Expression(G) = (x^2 + y^2/a^2 = 1);Coordinate(B) = (-1, 0);Eccentricity(G)=e;e = (2*sqrt(5))/5;PointOnCurve(P, G)", "coords": { "B": [ -1.0, 0.0 ] } }, { "index": 786, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0786.png", "answer": "{(2,3),(-2,3)}", "fact_expr": "C: Ellipse;l: Line;P: Point;B: Point;A: Point;Expression(C) = (x^2/16 + y^2/12 = 1);Coordinate(P) = (0, 6);PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(P,B)) = A", "coords": { "P": [ 0.0, 6.0 ] } }, { "index": 787, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0787.png", "answer": "y^2=12*x", "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;A: Point;B: Point;H:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(H) = (-1, 0);Focus(C) = F;PointOnCurve(H, G);Intersection(G, C) = {A, B};Min(4*Abs(LineSegmentOf(F, A)) + Abs(LineSegmentOf(F, B))) = 19", "coords": { "H": [ -1.0, 0.0 ] } }, { "index": 788, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0788.png", "answer": "sqrt(3)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;B: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(C)) =A;Intersection(l,RightPart(C))=B;IsDiameter(LineSegmentOf(A,B),G);PointOnCurve(F2,G);Abs(LineSegmentOf(A,F2))=Abs(LineSegmentOf(B,F2))", "coords": {} }, { "index": 789, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0789.png", "answer": "pm*sqrt(6)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;P: Point;PointOnCurve(P, C);Negation(P=A);Negation(P=B);M: Point;Intersection(LineOf(A, P), yAxis) = M;N: Point;PointOnCurve(A, LineOf(A, N));IsParallel(LineOf(A, N), LineOf(B, P));Intersection(LineOf(A, N), yAxis) = N;S: Point;t: Number;Coordinate(S) = (t, 0);DotProduct(VectorOf(S, M), VectorOf(S, N)) = 5", "coords": {} }, { "index": 790, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0790.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1,F2};PointOnCurve(P,G);AngleOf(F1,P,F2)=ApplyUnit(90,degree)", "coords": {} }, { "index": 791, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0791.png", "answer": "2/5", "fact_expr": "C: Ellipse;l: Line;l1:Line;A: Point;B: Point;D: Point;F: Point;Expression(C) = (x^2/25 + y^2/9 = 1);RightFocus(C) = F;PointOnCurve(F,l);Negation(IsParallel(l,yAxis));Intersection(l,C)={A,B};PerpendicularBisector(LineSegmentOf(A, B)) = l1;Intersection(MajorAxis(C),l1)=D", "coords": {} }, { "index": 792, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0792.png", "answer": "sqrt(3)", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(E) = F;F: Point;M: Point;N: Point;J: Line;OneOf(Asymptote(E)) = J;PointOnCurve(M, J) = True;PointOnCurve(N, J) = True;Abs(LineSegmentOf(F, M)) = Abs(LineSegmentOf(F, N));Abs(LineSegmentOf(F, N)) = sqrt(3)*a;DotProduct(VectorOf(F, M), VectorOf(F, N)) = a^2", "coords": {} }, { "index": 793, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0793.png", "answer": "sqrt(13)-2", "fact_expr": "C: Hyperbola;b: Number;a: Number;Q: Point;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);PointOnCurve(Q,Asymptote(C));Quadrant(P)=1;Quadrant(Q)=1;2*VectorOf(Q, P) = VectorOf(P, F2);DotProduct(VectorOf(Q, F1), VectorOf(Q, F2)) = 0", "coords": {} }, { "index": 794, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0794.png", "answer": "[0,2]", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/12 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 795, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0795.png", "answer": "0", "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;y0:Number;b>0;Expression(G) = (x^2/2 - y^2/b^2 = 1);Coordinate(P) = (sqrt(3), y0);LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G)))=(y=x);PointOnCurve(P, G)", "coords": {} }, { "index": 796, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0796.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 2", "coords": {} }, { "index": 797, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0797.png", "answer": "7", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Eccentricity(G) = 5/4;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(F1, P, F2)) = 9", "coords": {} }, { "index": 798, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0798.png", "answer": "11", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/12 = 1);P: Point;PointOnCurve(P, G);XCoordinate(P) = XCoordinate(RightFocus(G))", "coords": {} }, { "index": 799, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0799.png", "answer": "(1, 3]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Min(Abs(LineSegmentOf(P, F1))^2/Abs(LineSegmentOf(P, F2))) = 8*a;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 800, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0800.png", "answer": "pi/3", "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;Intersection(Directrix(G),xAxis)=M;PointOnCurve(N,G);Abs(LineSegmentOf(M,N))=2*Abs(LineSegmentOf(N,F))", "coords": {} }, { "index": 801, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0801.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F2,L);IsPerpendicular(Asymptote(C),L);Center(G)=F1;Radius(G)=Abs(LineSegmentOf(O,F1));IsTangent(L,G)", "coords": {} }, { "index": 802, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 803, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_0803.png", "answer": "2/7", "fact_expr": "C: Ellipse;a: Number;G: Line;a>sqrt(2);Expression(C) = (y^2/3 + x^2/a^2 = 1);Expression(G) = (y = 6*x);Eccentricity(C) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 804, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0804.png", "answer": "9", "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(A, G);Distance(A, Focus(G)) = 10", "coords": {} }, { "index": 805, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0805.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;Q: Point;F1: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;PointOnCurve(P,G);PointOnCurve(Q, RightDirectrix(G));VectorOf(P, Q) = 2*VectorOf(F1, O);lambda:Number;VectorOf(F1, Q) = lambda*(VectorOf(F1, O)/Abs(VectorOf(F1, O)) + VectorOf(F1, P)/Abs(VectorOf(F1, P)));lambda>0", "coords": {} }, { "index": 806, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0806.png", "answer": "sqrt(5)-1", "fact_expr": "G: Parabola;P: Point;A: Point;M: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2, 0);PointOnCurve(P,G);Projection(P, xAxis) = M", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 807, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0807.png", "answer": "8", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)/3;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) + Distance(P, F2) = 12", "coords": {} }, { "index": 808, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0808.png", "answer": "21", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/25 - y^2/24 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 11", "coords": {} }, { "index": 809, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0809.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 4/3);PointOnCurve(P,Asymptote(G))", "coords": { "P": [ 1.0, 1.3333333333333333 ] } }, { "index": 810, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0810.png", "answer": "2*sqrt(2)", "fact_expr": "E: Parabola;p: Number;A: Point;F: Point;O: Origin;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(A, E);Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(O, F));Abs(LineSegmentOf(O, A)) = sqrt(10)", "coords": {} }, { "index": 811, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 812, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 813, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0813.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "l: Line;C: Hyperbola;a: Number;A: Point;B: Point;F2: Point;F1: Point;a>0;Expression(C) = (-y^2/2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);IsPerpendicular(xAxis,l);Intersection(l, C) = {A, B};Area(TriangleOf(A, B, F2)) = 2*sqrt(6)", "coords": {} }, { "index": 814, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 815, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0815.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M: Point;PointOnCurve(M, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(M, F1) = 4", "coords": {} }, { "index": 816, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0816.png", "answer": "y^2 = 8*x", "fact_expr": "l: Line;G: Parabola;p: Number;H: Circle;O: Origin;F:Point;p>0;Expression(G) = (y^2 = 2*p*x);Directrix(G)=l;Radius(H)=3;Vertex(G)=O;Focus(G)=F;PointOnCurve(O,H);PointOnCurve(F,H);IsTangent(H,l)", "coords": {} }, { "index": 817, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 818, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0818.png", "answer": "sqrt(6)/3", "fact_expr": "M: Point;Coordinate(M) = (1, 1);Slope(l) = -1/3;PointOnCurve(M, l) = True;l: Line;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;B: Point;A: Point", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 819, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0819.png", "answer": "2", "fact_expr": "E: Parabola;p: Number;A: Point;F: Point;B: Point;G: Circle;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(A, E);Center(G) = A;IsTangent(G, yAxis);Intersection(G, LineSegmentOf(A, F)) = B;Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(B, F));Length(InterceptChord(PerpendicularBisector(LineSegmentOf(A, F)), G)) = sqrt(7)", "coords": {} }, { "index": 820, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0820.png", "answer": "{3,1/3}", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;Focus(C) = F;F: Point;Coordinate(F) = (1, 0);PointOnCurve(F, G) = True;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;Intersection(Directrix(C), xAxis) = M;M: Point;Area(TriangleOf(M, A, B)) = 8*sqrt(3)/3", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 821, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 822, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 823, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0823.png", "answer": "{ApplyUnit(45, degree), pi/4}", "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;K: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(M, C);Focus(C) = F;Abs(LineSegmentOf(M, F)) = p;Intersection(Directrix(C), xAxis) = K", "coords": {} }, { "index": 824, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0824.png", "answer": "1+2*sqrt(2)", "fact_expr": "l: Line;C: Hyperbola;P: Point;F1: Point;Q: Point;Expression(C) = (x^2/2 - y^2 = 1);PointOnCurve(P, RightPart(C));OneOf(Asymptote(C))=l;Projection(P,l)=Q;LeftFocus(C)=F1", "coords": {} }, { "index": 825, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0825.png", "answer": "[sqrt(2)/2,1)", "fact_expr": "G: Ellipse;m: Number;n: Number;H: Curve;e: Number;m > n;n > 0;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2 + y^2 = Abs(m - n));IsIntersect(G, H);Eccentricity(G) = e", "coords": {} }, { "index": 826, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0826.png", "answer": "3", "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 827, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 828, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 829, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 830, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0830.png", "answer": "x^2-y^2/24=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;O: Origin;Abs(LineSegmentOf(F1, F2)) = 10;M: Point;PointOnCurve(M, LeftPart(G)) = True;Abs(LineSegmentOf(O, M)) = sqrt(a^2 + b^2);4*Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 831, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0831.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;Distance(P, Directrix(G)) = d1;d1: Number;l: Line;Expression(l) = (4*x - 3*y + 16 = 0);Distance(P, l) = d2;d2: Number", "coords": {} }, { "index": 832, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0832.png", "answer": "(0,20)", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;I: Point;Expression(C) = (x^2/9 - y^2/16 = 1);LeftFocus(C)=F1;RightFocus(C) = F2;Quadrant(P) = 1;PointOnCurve(P, C);Incenter(TriangleOf(P, F1, F2)) = I", "coords": {} }, { "index": 833, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0833.png", "answer": "pm*2*sqrt(2)", "fact_expr": "l: Line;Expression(l)=(x=m*y+1);m: Number;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;F: Point;Focus(C)=F;PointOnCurve(F,l);A: Point;B: Point;Intersection(l,C)={A,B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 834, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0834.png", "answer": "16/3", "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B};Intersection(LineOf(A, B), Directrix(G)) = C;Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 835, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0835.png", "answer": "4*sqrt(5)", "fact_expr": "G: Hyperbola;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 - y^2/3 = 1);Expression(H) = (y = x + 1);Intersection(H, G) = {A,B}", "coords": {} }, { "index": 836, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0836.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "H: Ellipse;Expression(H) = (x^2/8 + y^2/2 = 1);G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 837, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 13.0, "b": 12.0, "major_axis": "x", "x_coef": 169.0, "y_coef": 144.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0837.png", "answer": "22", "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/169 + y^2/144 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 838, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0838.png", "answer": "y=2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)", "coords": {} }, { "index": 839, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0839.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 840, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0840.png", "answer": "x=-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 841, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0841.png", "answer": "", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (4*x - 3*y = 0)", "coords": {} }, { "index": 842, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0842.png", "answer": "sqrt(34)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = (3/5)*x)", "coords": {} }, { "index": 843, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 844, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0844.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "C:Ellipse;a:Number;b:Number;lambda:Number;F1:Point;P:Point ;Q:Point;O:Origin;a>b ;b>0;Expression(C)=(x^2/a^2+y^2/b^2=1);LeftFocus(C)=F1;PointOnCurve(P,C);PointOnCurve(Q,RightDirectrix(C));VectorOf(P,Q)=2*VectorOf(F1,O);VectorOf(F1,Q) = lambda * (VectorOf(F1,P)/Abs(VectorOf(F1,P))+VectorOf(F1,O)/Abs(VectorOf(F1,O)));lambda>0;Negation(RightVertex(C)=P)", "coords": {} }, { "index": 845, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/test/hyperbola/problem_0845.png", "answer": "y^2/3-x^2=1", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x);P: Point;Coordinate(P) = (sqrt(3), 2*sqrt(3));PointOnCurve(P, G) = True", "coords": { "P": [ 1.7320508075688772, 3.4641016151377544 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 846, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0846.png", "answer": "1", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Line;M: Point;F: Point;A: Point;p>0;x0: Number;x0 > p/2;Expression(C) = (y^2 = 2*(p*x));Expression(H) = (x = p/2);Coordinate(M) = (x0, 2*sqrt(2));Focus(C) = F;PointOnCurve(M, C);Center(G) = M;Intersection(G, LineSegmentOf(M, F)) = A;Length(InterceptChord(H, G)) = sqrt(3)*Abs(LineSegmentOf(M, A));Abs(LineSegmentOf(M, A)) = 2", "coords": {} }, { "index": 847, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0847.png", "answer": "-1/3", "fact_expr": "G: Hyperbola;m: Real;n: Real;H: Ellipse;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2/5 + y^2/4 = 1);Eccentricity(G)=2;Focus(G)=Focus(H)", "coords": {} }, { "index": 848, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0848.png", "answer": "{(1,-4),(1,4)}", "fact_expr": "M: Parabola;P: Point;F: Point;Expression(M) = (y^2 = 16*x);Focus(M)=F;PointOnCurve(P, M);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 849, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0849.png", "answer": "2\ny=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)", "coords": {} }, { "index": 850, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0850.png", "answer": "x^2/9-y^2/16=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (5, 0);OneOf(Focus(G))=H;Expression(OneOf(Asymptote(G)))=(3*y-4*x=0)", "coords": { "H": [ 5.0, 0.0 ] } }, { "index": 851, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0851.png", "answer": "20", "fact_expr": "l1: Line;Expression(l1) = (2*x -y + 6 = 0);l2: Line;Expression(l2) = (x = -1);C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);WhenMin(Distance(P, l1) + Distance(P, l2))", "coords": {} }, { "index": 852, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0852.png", "answer": "y = pm*(4/3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;I: Point;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 20*x);Coordinate(I) = (15/4, 3);Focus(H) = F;RightFocus(G) = F;PointOnCurve(I, G)", "coords": { "I": [ 3.75, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 853, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0853.png", "answer": "10", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(P, G);Distance(P, yAxis) = 6", "coords": {} }, { "index": 854, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 855, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0855.png", "answer": "x=-1\n(1,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 856, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 857, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0857.png", "answer": "6-2*sqrt(6)", "fact_expr": "G: Ellipse;H: Circle;P: Point;A: Point;Q: Point;F: Point;Expression(G) = (x^2/6 + y^2/2 = 1);Expression(H) = (x^2 + y^2 = 8);RightFocus(G) = F;UpperVertex(G) = A;PointOnCurve(P, H);PointOnCurve(Q, G)", "coords": {} }, { "index": 858, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0858.png", "answer": "[0,sqrt(2)]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);P: Point;PointOnCurve(P,C) = True;DotProduct(VectorOf(O,P),VectorOf(P,F2)) >= -1;O: Origin;F2: Point;RightFocus(C) = F2", "coords": {} }, { "index": 859, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0859.png", "answer": "sqrt(2)\n0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G))) = (y = x);y0: Number;P: Point;Coordinate(P) = (sqrt(3), y0);PointOnCurve(P, G)", "coords": {} }, { "index": 860, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0860.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: LineSegment;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (3, 1);IsChordOf(H,G);MidPoint(H)=P;Slope(OverlappingLine(H))=2", "coords": { "P": [ 3.0, 1.0 ] } }, { "index": 861, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 862, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0862.png", "answer": "y=pm*(sqrt(10)/4)*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;RightFocus(G) = F;I: Point;Coordinate(I) = ((3*a^2)/p, (2*b^2)/p);PointOnCurve(I, G)", "coords": {} }, { "index": 863, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 864, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0864.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;K: Point;Coordinate(K) = (2, 0);H: Line;PointOnCurve(K, H);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;Intersection(H, Directrix(G)) = C;Area(TriangleOf(A, C, F))/Area(TriangleOf(B, C, F)) = 2/5", "coords": { "K": [ 2.0, 0.0 ] } }, { "index": 865, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0865.png", "answer": "sqrt(2)+1", "fact_expr": "B: Point;F: Point;C: Hyperbola;a: Number;b: Number;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C)=F;Intersection(LeftDirectrix(C),xAxis)=B;MidPoint(LineSegmentOf(B, F)) = A", "coords": {} }, { "index": 866, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0866.png", "answer": "[1/2,1)", "fact_expr": "A: Point;P: Point;G: Ellipse;b: Number;a: Number;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Intersection(RightDirectrix(G), xAxis) = A;PointOnCurve(P,G);PointOnCurve(F,PerpendicularBisector(LineSegmentOf(A,P)))", "coords": {} }, { "index": 867, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 868, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0868.png", "answer": "4", "fact_expr": "F1:Point;F2:Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;b: Number;a: Number;a>0;b>0;e:Number;Eccentricity(C)=e;P: Point;PointOnCurve(P,RightPart(C));DotProduct(VectorOf(P,F2),VectorOf(F1,F2)) =0;Abs(VectorOf(P, F2)) = (2*sqrt(3))/3;e=2*sqrt(3)/3", "coords": {} }, { "index": 869, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0869.png", "answer": "2+sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Negation(M=N);F: Point;Focus(G) = F;AngleOf(M, F, N) = ApplyUnit(135, degree);IsChordOf(LineSegmentOf(M, N), G);P: Point;MidPoint(LineSegmentOf(M, N)) = P;l: Line;Expression(l) = (y=-1/16);d: Number;Distance(P, l) = d;lambda: Number;Abs(LineSegmentOf(M, N))^2 = (d^2)*lambda", "coords": {} }, { "index": 870, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0870.png", "answer": "{[-1-sqrt(3),0)+(0,1+sqrt(3)]}", "fact_expr": "l: Line;C: Curve;M: Point;Q: Point;N: Point;n:Number;Expression(C) = (x^2/2 + y^2 = 1);Expression(l)=(y=1);Intersection(l,yAxis)=M;PointOnCurve(Q,l);Negation(Q=M);XCoordinate(MidPoint(Q)) = n;PointOnCurve(N, C);AngleOf(M, Q, N) = ApplyUnit(45, degree)", "coords": {} }, { "index": 871, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0871.png", "answer": "sqrt(3) + 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, RightPart(G));DotProduct((VectorOf(O, F2) + VectorOf(O, M)),VectorOf(F2, M)) = 0;Abs(VectorOf(M, F1)) = sqrt(3)*Abs(VectorOf(M, F2))", "coords": {} }, { "index": 872, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, 1.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0872.png", "answer": "16", "fact_expr": "H: Line;b: Number;a: Number;c: Number;b*c>0;Expression(H) = (a*x + b*y + c - 1 = 0);G: Circle;Expression(G) = (-2*y + x^2 + y^2 - 15 = 0);PointOnCurve(Center(G),H) = True", "coords": {} }, { "index": 873, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 874, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_0874.png", "answer": "(pm*2, 0)\ny=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 875, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 876, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0876.png", "answer": "[-1/3, 0)+(0, 1/3]", "fact_expr": "G: Hyperbola;k: Real;H: Circle;Expression(G) = (x^2/9 - y^2/4 = k^2);Expression(H) = (x^2 + y^2 = 1);IsIntersect(G, H)", "coords": {} }, { "index": 877, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 5.291502622129181, "major_axis": "x", "x_coef": 64.0, "y_coef": 28.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0877.png", "answer": "5", "fact_expr": "C: Ellipse;P: Point;F1: Point;O: Origin;M: Point;Expression(C) = (x^2/64 + y^2/28 = 1);LeftFocus(C) = F1;PointOnCurve(P, C);Distance(P,LeftFocus(C))=6;MidPoint(LineSegmentOf(P,F1)) = M", "coords": {} }, { "index": 878, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0878.png", "answer": "ApplyUnit(90,degree)", "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/2 - y^2/2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);OneOf(Intersection(G,H))=P;Focus(G)={F1,F2};Focus(H)={F1,F2}", "coords": {} }, { "index": 879, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0879.png", "answer": "(-2, 0)+(0, 2)", "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 = R^2);R: Number;G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1);NumIntersection(H, G) = 0", "coords": {} }, { "index": 880, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0880.png", "answer": "x=-3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x)", "coords": {} }, { "index": 881, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0881.png", "answer": "sqrt(5)-2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;LeftFocus(G) = F1;H: Circle;Expression(H) = (x^2 + y^2 = 4);L1: Line;TangentOfPoint(F1, H) = L1;Intersection(L1, RightPart(G)) = P;P: Point;TangentPoint(L1, H) = T;T: Point;MidPoint(LineSegmentOf(P, F1)) = M;M: Point;O: Origin", "coords": {} }, { "index": 882, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0882.png", "answer": "y=-1/2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y)", "coords": {} }, { "index": 883, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0883.png", "answer": "[sqrt(2), +oo)", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;Center(InscribedCircle(TriangleOf(P, F1, F2))) = I;Area(TriangleOf(I, P, F1)) - Area(TriangleOf(I, P, F2)) <= (sqrt(2)/2)*Area(TriangleOf(I, F1, F2))", "coords": {} }, { "index": 884, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0884.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/m = 1);m: Real;Coordinate(OneOf(Focus(G))) = (0, 2)", "coords": {} }, { "index": 885, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.552875593585339, "b": 2.3303634978370953, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3057248743840209e-06, "converged": true, "iterations": 98 }, "output_path": "results/test/hyperbola/problem_0885.png", "answer": "x^2/4-y^2/9=1", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/2)*x)", "coords": {} }, { "index": 886, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0886.png", "answer": "4/5", "fact_expr": "alpha: Number;Inclination(l) = alpha;l: Line;Slope(l) = Eccentricity(G);G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 887, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0887.png", "answer": "3", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 5/2", "coords": {} }, { "index": 888, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0888.png", "answer": "12", "fact_expr": "C1: Ellipse;C2: Ellipse;O: Origin;M: Point;N: Point;Expression(C1) = (x^2/16+y^2/9=1);Expression(C2) = (x^2/64+y^2/36=1);PointOnCurve(M,C1);PointOnCurve(N,C2)", "coords": {} }, { "index": 889, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0889.png", "answer": "(1,sqrt(5))", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;PointOnCurve(RightFocus(E), G);Slope(G) = 2;NumIntersection(G, RightPart(E)) = 2", "coords": {} }, { "index": 890, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_0890.png", "answer": "x^2/4-y^2/12 = 1", "fact_expr": "E: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(E) = (x^2/a^2 - y^2/b^2 = 1);Expression(OneOf(Asymptote(E))) = (y = sqrt(3)*x);F: Parabola;Expression(F) = (y^2 = 16*x);OneOf(Focus(E)) = Focus(F)", "coords": {} }, { "index": 891, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0891.png", "answer": "1/4", "fact_expr": "C: Parabola;l1:Line;l2:Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1,C)={A, B};Intersection(l2,C)={D, E}", "coords": {} }, { "index": 892, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_0892.png", "answer": "12", "fact_expr": "G: Parabola;H: Ellipse;m: Number;n: Number;Expression(G) = (y^2 = 4*x);m > n;n > 0;Expression(H) = (y^2/n + x^2/m = 1);Eccentricity(H) = 1/2;OneOf(Focus(H))=Focus(G)", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 893, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0893.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);P: Point;Intersection(H, G) = P;LineSegmentOf(P, O) = 3*sqrt(5)", "coords": {} }, { "index": 894, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0894.png", "answer": "y = pm*2*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/16 = 1)", "coords": {} }, { "index": 895, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/test/hyperbola/problem_0895.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x);Expression(RightDirectrix(G)) = (x = sqrt(5)/5)", "coords": {} }, { "index": 896, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_0896.png", "answer": "2", "fact_expr": "C: Hyperbola;m: Number;Expression(C) = (-y^2/4 + x^2/m = 1);Eccentricity(C) = sqrt(5)", "coords": {} }, { "index": 897, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0897.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;OneOf(Focus(C)) = F;G: Circle;Expression(G) = (x^2 + y^2 = a^2);TangentOfPoint(F, G) = {l1, l2};l1: Line;l2: Line;TangentPoint(l1,G) = A;TangentPoint(l2,G) = B;A: Point;B: Point;AngleOf(A, F, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 898, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0898.png", "answer": "(1,3]", "fact_expr": "F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P,LeftPart(G)) = True;Abs(LineSegmentOf(P,F2))^2/Abs(LineSegmentOf(P,F1)) = 8*a", "coords": {} }, { "index": 899, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 900, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0900.png", "answer": "2", "fact_expr": "G: Ellipse;m: Number;m>0;Expression(G) = (y^2 + x^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 2", "coords": {} }, { "index": 901, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0901.png", "answer": "5/4", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*(3/4)*x)", "coords": {} }, { "index": 902, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0902.png", "answer": "[1, 4]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MinorAxis(G)) = 2;A: Point;UpperVertex(G) = A;B: Point;LeftVertex(G) = B;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(F1, A, B)) = (2 - sqrt(3))/2;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 903, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0903.png", "answer": "4", "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F: Point;Coordinate(F) = (3, 0);Abs(LineSegmentOf(P, F)) = 2;O: Origin;M: Point;MidPoint(LineSegmentOf(P, F)) = M", "coords": { "F": [ 3.0, 0.0 ] } }, { "index": 904, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 905, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0905.png", "answer": "0", "fact_expr": "C: Parabola;p:Number;p>0;A: Point;B: Point;P: Point;F: Point;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(F,LineSegmentOf(A,B));l1:Line;l2:Line;TangentOfPoint(A,C)=l1;TangentOfPoint(B,C)=l2;Intersection(l1,l2)=P", "coords": {} }, { "index": 906, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0906.png", "answer": "2*sqrt(3)", "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;B: Point;M: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (4, 0);DotProduct(VectorOf(O, A), VectorOf(O, B)) = -4;Intersection(l,C)={A,B};Quadrant(A) = 1;WhenMin(Area(TriangleOf(A,B,M)))", "coords": { "M": [ 4.0, 0.0 ] } }, { "index": 907, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0907.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;G: Line;a: Number;b: Number;P: Point;F1: Point;F2: Point;Q: Point;Expression(G) = (y = x*(b/a));LeftFocus(C) = F1;RightFocus(C) = F2;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));MidPoint(LineSegmentOf(P, F1)) = Q;PointOnCurve(Q, G);a > 0;b > 0", "coords": {} }, { "index": 908, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0908.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 909, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0909.png", "answer": "3*x-2*y-16=0", "fact_expr": "G: Hyperbola;H: Line;B: Point;C: Point;A: Point;Expression(G) = (x^2 - 4*y^2 = 16);Coordinate(A) = (6, 1);PointOnCurve(A, H);Intersection(H, G) = {B, C};MidPoint(LineSegmentOf(B,C))=A", "coords": { "A": [ 6.0, 1.0 ] } }, { "index": 910, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0910.png", "answer": "2*sqrt(2)-2", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 911, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 912, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_0912.png", "answer": "x^2-y^2/4=1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (sqrt(2), 2);PointOnCurve(H,G) = True;Expression(Asymptote(G)) = (y = pm*2*x)", "coords": { "H": [ 1.4142135623730951, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 913, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 914, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0914.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;A: Point;M: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2, 0);PointOnCurve(M, G)", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 915, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0915.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) = 6", "coords": {} }, { "index": 916, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0916.png", "answer": "2", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);M: Point;Coordinate(M) = (2, 0);l: Line;B: Point;A: Point;PointOnCurve(M,l) = True;Intersection(l, E) = {A, B};C: Point;Intersection(Directrix(E) , l) = C;Quadrant(A) = 1 ;Abs(LineSegmentOf(C, B))/Abs(LineSegmentOf(A, B)) = 2/3", "coords": { "M": [ 2.0, 0.0 ] } }, { "index": 917, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0917.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 918, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 919, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0919.png", "answer": "9*sqrt(2)/4", "fact_expr": "C: Hyperbola;a, b: Number;b > a;a > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;Coordinate(P) = (2, sqrt(2));PointOnCurve(P, C);Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2)) = 3;A: Point;Intersection(LineSegmentOf(P, F1), C) = {P, A}", "coords": { "P": [ 2.0, 1.4142135623730951 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 920, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 921, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0921.png", "answer": "(1,sqrt(2)+1)", "fact_expr": "G: Hyperbola;a: Number;b: Number;c:Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Sin(AngleOf(P, F1, F2))/Sin(AngleOf(P, F2, F1)) = a/c", "coords": {} }, { "index": 922, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0922.png", "answer": "4", "fact_expr": "l: Line;C: Hyperbola;A: Point;F1: Point;B: Point;F2: Point;Expression(C) = (x^2 - y^2/8 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F2, l);Intersection(l, LeftPart(C)) = A;Abs(LineSegmentOf(A, F1)) = Abs(LineSegmentOf(B, F1));Intersection(l, RightPart(C)) = B", "coords": {} }, { "index": 923, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0923.png", "answer": "y^2=16*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C) = True;Abs(LineSegmentOf(M, F)) = 5;IsDiameter(LineSegmentOf(M, F), G) = True;G: Circle;PointOnCurve(D, G) = True;D: Point;Coordinate(D) = (0,2);p > Abs(LineSegmentOf(M, F))", "coords": { "D": [ 0.0, 2.0 ] } }, { "index": 924, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0924.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);M: Point;Coordinate(M) = (-1, 1/2);H: LineSegment;IsChordOf(H, G) = True;MidPoint(H) = M", "coords": { "M": [ -1.0, 0.5 ] } }, { "index": 925, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_0925.png", "answer": "16", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Line;H: Point;A: Point;B: Point;F2: Point;F1:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (-2, 0);PointOnCurve(H, C);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {A, B};AngleOf(A, F2, B) = ApplyUnit(45, degree);Abs(LineSegmentOf(A, F2)) = 8", "coords": { "H": [ -2.0, 0.0 ] } }, { "index": 926, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0926.png", "answer": "(sqrt(7) - 1)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G) = A;Z: Line;PointOnCurve(F, Z);IsPerpendicular(Z, xAxis);Intersection(Z, G) = P;Radius(InscribedCircle(TriangleOf(A, F, P))) = sqrt(a^2 - b^2)/2", "coords": {} }, { "index": 927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0927.png", "answer": "x^2/4-y^2/5=1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = (sqrt(5)/2)*x);G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);Focus(C) = Focus(G)", "coords": {} }, { "index": 928, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 929, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0929.png", "answer": "36", "fact_expr": "C: Ellipse;a: Number;F1: Point;P: Point;F2: Point;a>3;Expression(C) = (y^2/9 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 930, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.6638543605804443, "b": 2.2922656641629255, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.1016254280344046e-06, "converged": true, "iterations": 106 }, "output_path": "results/test/ellipse/problem_0930.png", "answer": "(-1,1)", "fact_expr": "G: Ellipse;n: Number;m: Number;P: Point;A: Point;B: Point;m>0;n>0;Expression(G) = (y^2/n + x^2/m = 1);Coordinate(A) = (-1, -3);Coordinate(B) = (2, 0);Coordinate(P) = (x1, y1);x1:Number;y1:Number;-10;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 935, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0935.png", "answer": "5*x+y-1=0", "fact_expr": "l: Line;Expression(l) = (k*x - y + 1 = 0);k: Number;C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;P: Point;Intersection(Directrix(C), xAxis) = P;DotProduct(VectorOf(O, A), VectorOf(O, B)) < 0;WhenMin(DotProduct(VectorOf(P, A), VectorOf(P, B)))", "coords": {} }, { "index": 936, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0936.png", "answer": "sqrt(13)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};H: Parabola;Expression(H) = (y^2 = 4*x);A: Point;Intersection(Z1, Directrix(H)) = A;B: Point;Intersection(Z2, Directrix(H)) = B;O: Origin;Area(TriangleOf(A, O, B)) = 2*sqrt(3);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 937, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0937.png", "answer": "4", "fact_expr": "G: Parabola;a: Number;Expression(G) = (y^2 = 2*(a*x));Expression(Directrix(G)) = (x = -2)", "coords": {} }, { "index": 938, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_0938.png", "answer": "(y^2=2*x+1)&(-1/2<=x<=1/2)", "fact_expr": "C: Circle;P: Point;Q: Point;x0:Number;y0:Number;Expression(C) = (x^2 + y^2 = 1);Coordinate(P) = (x0, y0);Coordinate(Q) = (x0*y0, x0 + y0);PointOnCurve(P, C)", "coords": {} }, { "index": 939, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0939.png", "answer": "2*pi/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);p + x1 + x2 = ((2*sqrt(3))/3)*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 940, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0940.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;M: Point;E: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;L:Line;PointOnCurve(F,L);IsPerpendicular(L,Asymptote(G));FootPoint(L,Asymptote(G))=M;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = E;LineSegmentOf(F, M) = LineSegmentOf(M, E)", "coords": {} }, { "index": 941, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 942, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0942.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 943, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0943.png", "answer": "8*sqrt(5)/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);F1: Point;LeftFocus(G) = F1;F2: Point;RightFocus(G) = F2;L: Line;PointOnCurve(F1, L)= True;IsPerpendicular(L, xAxis) = True;Intersection(L, G) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 944, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0944.png", "answer": "(-1+sqrt(5))/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;O: Origin;A: Point;B: Point;F: Point;a > b;b > 0;RightFocus(G) = F;L:Line;IsPerpendicular(L,xAxis);PointOnCurve(F,L);Intersection(L, G) = {A, B};DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0", "coords": {} }, { "index": 945, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0945.png", "answer": "[2,+oo)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, C) = True;L1: Line;L2: Line;Asymptote(C) = {L1,L2};Distance(P,L1) * Distance(P,L2) = 1;c: Number;HalfFocalLength(C) = c", "coords": {} }, { "index": 946, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0946.png", "answer": "10", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;Expression(C) = (y^2 = 2*p*x);Expression(l)=(y=2*x-4);Focus(C)=F;PointOnCurve(F,l);Intersection(l,C)={A,B};F:Point", "coords": {} }, { "index": 947, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0947.png", "answer": "x^2/4-y^2/12=1", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);G: Hyperbola;Focus(H) = Focus(G);Eccentricity(G) = 2", "coords": {} }, { "index": 948, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0948.png", "answer": "16/3", "fact_expr": "C1: Hyperbola;P: Point;F: Point;Expression(C1) = (x^2/4 - y^2/5 = 1);PointOnCurve(P, C1);RightFocus(C1) = F;LineSegmentOf(P, F) = 4", "coords": {} }, { "index": 949, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0949.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;b/a = 2;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 950, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0950.png", "answer": "y = pm*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);AngleOf(P, B, A) = pi/2 + AngleOf(P, A, B)", "coords": {} }, { "index": 951, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0951.png", "answer": "sqrt(3)*x + pm*y=0", "fact_expr": "P: Point;Coordinate(P) = (-2, 0);PointOnCurve(P, C);C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(C) = Focus(G)", "coords": { "P": [ -2.0, 0.0 ] } }, { "index": 952, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0952.png", "answer": "y = -1/2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y)", "coords": {} }, { "index": 953, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0953.png", "answer": "1", "fact_expr": "C1: Hyperbola;C: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Expression(C1)=(2*x^2/a^2-y^2=1);Focus(C)=Focus(C1);OneOf(Intersection(C,C1))=M;Abs(LineSegmentOf(M,F1))*Abs(LineSegmentOf(M,F2))=2", "coords": {} }, { "index": 954, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0954.png", "answer": "4*sqrt(3)/3", "fact_expr": "H: Line;Expression(H) = (y = sqrt(3)*(x - 1));G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin", "coords": {} }, { "index": 955, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0955.png", "answer": "sqrt(5) - 1", "fact_expr": "l: Line;G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (2*x - y + 3 = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 956, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/test/hyperbola/problem_0956.png", "answer": "y = pm*(sqrt(3)/2)*x", "fact_expr": "C1:Hyperbola;C2:Hyperbola;P:Point;Q:Point;XCoordinate(P)=XCoordinate(Q);YCoordinate(P)=2*YCoordinate(Q);Locus(P)=C1;Locus(Q)=C2;Expression(Asymptote(C1)) = (y = pm*(sqrt(3)*x))", "coords": {} }, { "index": 957, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0957.png", "answer": "2*sqrt(21)", "fact_expr": "C: Parabola;G: Circle;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Center(G) = F;Intersection(G, C) = {A, B};Intersection(G, Directrix(C)) = {D, E};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 958, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0958.png", "answer": "(9/2)*sqrt(2)", "fact_expr": "G: Parabola;l1: Line;l2: Line;P:Point;Expression(G) = (x^2 = 16*y);Expression(l1) = (x - y - 5 = 0);Expression(l2) = (y=-4);PointOnCurve(P,G)", "coords": {} }, { "index": 959, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0959.png", "answer": "5/3", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/16 + x^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (5, 0)", "coords": {} }, { "index": 960, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0960.png", "answer": "5", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 5", "coords": {} }, { "index": 961, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0961.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)", "coords": {} }, { "index": 962, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0962.png", "answer": "y^2=10*x", "fact_expr": "G: Parabola;p: Number;H: Line;O:Origin;p>0;Expression(G) = (y^2 = 2*(p*x));L:Line;PointOnCurve(Focus(G),H);PointOnCurve(O,L);IsPerpendicular(H,L);Coordinate(FootPoint(H,L))=(2,1)", "coords": {} }, { "index": 963, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/test/hyperbola/problem_0963.png", "answer": "y^2/3-x^2=1", "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G),yAxis);FocalLength(G) = 4;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 964, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_0964.png", "answer": "{4,1/4}", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b + x^2/a = 1);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 965, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0965.png", "answer": "{x=0, y=-3, x+3*y+9=0}", "fact_expr": "H: Point;Coordinate(H) = (0, -3);l: Line;PointOnCurve(H, l);G: Parabola;Expression(G) = (y^2 = 4*x);NumIntersection(l, G) = 1", "coords": { "H": [ 0.0, -3.0 ] } }, { "index": 966, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 967, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_0967.png", "answer": "3/2", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 968, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0968.png", "answer": "8/3", "fact_expr": "G: Hyperbola;P: Point;A: Point;B: Point;O: Origin;m: Number;n: Number;Expression(G) = (x^2 - y^2/4 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A = B);Negation(A = P);Negation(P = B);VectorOf(P, A) + VectorOf(P, B) = 2*VectorOf(P, O);Slope(LineSegmentOf(P, A)) = m;Slope(LineSegmentOf(P, B)) = n", "coords": {} }, { "index": 969, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0969.png", "answer": "sqrt(3)/2", "fact_expr": "G: Hyperbola;H: Ellipse;a2: Number;b2: Number;a1: Number;b1: Number;F1: Point;P: Point;F2: Point;a2>0;b2>0;Expression(G) = (-y^2/b2^2 + x^2/a2^2 = 1);a1>b1;b1>0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);OneOf(Intersection(H, G)) = P;Focus(G) = {F1, F2};Focus(H) = {F1, F2};Focus(G) = Focus(H);e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 970, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/test/hyperbola/problem_0970.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 2);Expression(Asymptote(G)) = (y = pm*sqrt(2)*x);PointOnCurve(H,G)", "coords": { "H": [ 1.0, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 971, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0971.png", "answer": "-5", "fact_expr": "G: Hyperbola;H: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a1:Number;b1:Number;a1 > b1;b1 > 0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);Vertex(G)={A,B};In(A,Vertex(H));In(B,Vertex(H));PointOnCurve(P,G);PointOnCurve(M,H);Negation(P=A);Negation(P=B);Negation(M=A);Negation(M=B);LineSegmentOf(A, P) + LineSegmentOf(B, P) = lambda*(LineSegmentOf(A, M) + LineSegmentOf(B, M));lambda:Real;Slope(LineOf(A,P))=k1;Slope(LineOf(B,P))=k2;Slope(LineOf(A,M))=k3;Slope(LineOf(B,M))=k4;k1:Number;k2:Number;k3:Number;k4:Number;k1+k2=5", "coords": {} }, { "index": 972, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0972.png", "answer": "sqrt(3)-1", "fact_expr": "r: Ellipse;b: Number;a: Number;c: Number;M: Point;F1: Point;F2: Point;G:Line;a > b;b > 0;Expression(r) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*(c + x));LeftFocus(r)=F1;RightFocus(r)=F2;FocalLength(r)=2*c;OneOf(Intersection(G,r))=M;AngleOf(M,F1,F2)=2*AngleOf(M,F2,F1)", "coords": {} }, { "index": 973, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0973.png", "answer": "20", "fact_expr": "O: Origin;G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;Area(TriangleOf(O, P, F)) = 32", "coords": {} }, { "index": 974, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0974.png", "answer": "8", "fact_expr": "C: Parabola;A: Point;F: Point;K: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 975, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0975.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;K: Point;PointOnCurve(K, C);F: Point;Focus(C) = F;G: Point;Coordinate(G) = (4, 0);Min(Distance(K, F) + Distance(K, G)) = 5", "coords": { "G": [ 4.0, 0.0 ] } }, { "index": 976, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 977, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 978, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0978.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B}", "coords": {} }, { "index": 979, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 980, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0980.png", "answer": "-4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);IsChordOf(LineSegmentOf(A,B),G) = True;PointOnCurve(Focus(G),LineSegmentOf(A,B)) = True;x1: Number;y1: Number;x2: Number;y2: Number", "coords": {} }, { "index": 981, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0981.png", "answer": "(2*sqrt(6))/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);H: Line;Slope(H) = 1;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 982, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0982.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;k:Number;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)/2;RightFocus(C)=F;PointOnCurve(F, G);Slope(G)=k;k>0;Intersection(G, C) = {A,B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 983, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0983.png", "answer": "12", "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/m = 1);e: Number;Eccentricity(G) = e;e = 1/2;m: Real;PointOnCurve(Focus(G), xAxis)", "coords": {} }, { "index": 984, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0984.png", "answer": "2*sqrt(3)", "fact_expr": "C: Hyperbola;G: Line;F: Point;P: Point;Q: Point;Expression(C) = (x^2/4 - y^2/12 = 1);Expression(G) = (x = 2);RightFocus(C) = F;Quadrant(P) = 1;PointOnCurve(P, G);Quadrant(Q) = 1;Intersection(LineOf(P, F), OneOf(Asymptote(C))) = Q;VectorOf(F, P) = 2*VectorOf(P, Q)", "coords": {} }, { "index": 985, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 986, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0986.png", "answer": "sqrt(3)/2", "fact_expr": "G: Parabola;a: Number;M: Ellipse;b: Number;H: Line;B: Point;A: Point;P: Point;Expression(G) = (y^2 = -a*x);a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = -x);RightVertex(M) = A;UpperVertex(M) = B;Intersection(LineOf(A,B), H) = P;PointOnCurve(P, G)", "coords": {} }, { "index": 987, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0987.png", "answer": "2*sqrt(3)", "fact_expr": "A: Point;C: Point;N: Point;M: Point;Coordinate(A) = (0, 1);Coordinate(C) = (0, 5);E: Curve;Expression(E) = (x^2 = 4*y);Locus(M) = E;Abs(LineSegmentOf(N, C)) = 2", "coords": { "A": [ 0.0, 1.0 ], "C": [ 0.0, 5.0 ] } }, { "index": 988, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0988.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l) = True;Intersection(l,G) = {A,B};B: Point;A: Point;P: Point;Coordinate(P) = (-5/3, 0);IsPerpendicular(LineSegmentOf(P, B), LineSegmentOf(A, B))", "coords": { "P": [ -1.6666666666666667, 0.0 ] } }, { "index": 989, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0989.png", "answer": "x^2+(y-2)^2=16", "fact_expr": "G: Parabola;H: Circle;L: Line;P:Point;D:Point;Expression(G) = (x^2 = 8*y);Expression(L) = (y = 2);Intersection(G,L)={P,D};PointOnCurve(P,H);PointOnCurve(D,H);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 990, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0990.png", "answer": "3/4", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;A:Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F;RightVertex(G)=A;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F),xAxis);Abs(LineSegmentOf(P,F))=(1/4)*Abs(LineSegmentOf(A,F))", "coords": {} }, { "index": 991, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0991.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 1;O: Origin", "coords": {} }, { "index": 992, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_0992.png", "answer": "12", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/8 + y^2/m = 1);OverlappingLine(MajorAxis(G),yAxis);FocalLength(G) = 4", "coords": {} }, { "index": 993, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_0993.png", "answer": "{2,6}", "fact_expr": "G: Parabola;p: Real;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (m, 2*sqrt(3));PointOnCurve(A, G);Distance(A, Focus(G)) = 4", "coords": {} }, { "index": 994, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 995, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0995.png", "answer": "(5/3,2*sqrt(3)/3)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 996, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_0996.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F: Point;N: Point;M:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;l1:Line;l2:Line;Expression(l1)=(y=(b/a)*x);Expression(l2)=(y=-(b/a)*x);L:Line;PointOnCurve(F,L);IsPerpendicular(l1,L);FootPoint(l1,L)=M;Intersection(L,l2)=N;Abs(LineSegmentOf(O, F)) = Abs(LineSegmentOf(F, N));Asymptote(C)={l1,l2}", "coords": {} }, { "index": 997, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 998, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 999, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_0999.png", "answer": "1/2", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;Expression(C) = (y = x^2/4);Coordinate(M) = (1, -1);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(A, M), VectorOf(B, M)) = 0;k:Number", "coords": { "M": [ 1.0, -1.0 ] } }, { "index": 1000, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1000.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;M: Point;x1:Number;y1:Number;Expression(G) = (x^2 = 4*y);Coordinate(M) = (x1, y1);PointOnCurve(M, G);Distance(M, Focus(G)) = 3;O:Origin", "coords": {} }, { "index": 1001, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1001.png", "answer": "x=2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)", "coords": {} }, { "index": 1002, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1002.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/6 - y^2/3 = 1)", "coords": {} }, { "index": 1003, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1003.png", "answer": "3", "fact_expr": "C: Parabola;A: Point;B: Point;D: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Coordinate(D) = (m, 0);x1:Number;x2:Number;y1:Number;y2:Number;m:Number;PointOnCurve(A, C);PointOnCurve(B, C);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = D;x1 + x2 = 2", "coords": {} }, { "index": 1004, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1004.png", "answer": "(x-2)^2+y^2=1", "fact_expr": "G: Circle;A: Point;P: Point;O: Origin;B: Point;x1:Number;y1:Number;Expression(G) = (x^2 + y^2 = 4);Coordinate(A) = (4, 0);Coordinate(P) = (x1, y1);PointOnCurve(B, G);VectorOf(O, A) + VectorOf(O, B) = 2*VectorOf(O, P)", "coords": {} }, { "index": 1005, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1005.png", "answer": "-9", "fact_expr": "G: Hyperbola;k: Real;Expression(G) = (x^2/4 + y^2/k = 1);Distance(OneOf(Focus(G)),Asymptote(G))=3", "coords": {} }, { "index": 1006, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/test/hyperbola/problem_1006.png", "answer": "pm*1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;m: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = m);l1: Line;l2: Line;Asymptote(C) = {l1, l2} ;Intersection(G, l1) = A;Intersection(G, l2) = B;Area(TriangleOf(A, O, B)) = sqrt(2);Eccentricity(C) = sqrt(3)", "coords": {} }, { "index": 1007, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1008, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1008.png", "answer": "25", "fact_expr": "C: Ellipse;Expression(C) = (y^2/9 + x^2/a = 1);a: Number;FocalLength(C) = 8", "coords": {} }, { "index": 1009, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1009.png", "answer": "x^2/6-y^2/2=1", "fact_expr": "G: Hyperbola;C: Hyperbola;H: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(H) = (3, -1);PointOnCurve(H, C);Asymptote(G) = Asymptote(C)", "coords": { "H": [ 3.0, -1.0 ] } }, { "index": 1010, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1011, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1011.png", "answer": "18+2*sqrt(17)", "fact_expr": "P: Point;Coordinate(P) = (0, 2);G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1);A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;PointOnCurve(A, G);PointOnCurve(B, G);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);lambda: Real;VectorOf(A, P) = lambda*VectorOf(P, B)", "coords": { "P": [ 0.0, 2.0 ] } }, { "index": 1012, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1012.png", "answer": "16", "fact_expr": "G: Ellipse;A: Point;B: Point;C: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(A) = (-3, 0);Coordinate(B) = (3, 0);PointOnCurve(C, G)", "coords": { "A": [ -3.0, 0.0 ], "B": [ 3.0, 0.0 ] } }, { "index": 1013, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1013.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;M: Point;PointOnCurve(M, G);Negation(Vertex(G) = M);O: Origin;VectorOf(O, M) = 2*VectorOf(O, N);N: Point;P: Point;H: Line;PointOnCurve(N,H);IsPerpendicular(LineOf(O,M),H);Intersection(H,xAxis) = P", "coords": {} }, { "index": 1014, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1014.png", "answer": "1", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C)", "coords": {} }, { "index": 1015, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1015.png", "answer": "3/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 1016, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1016.png", "answer": "(0,sqrt(5)/5]", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;G: Circle;A: Point;B: Point;a > b;b > 0;l1:Line;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(C),l1);IsPerpendicular(l1,xAxis);Intersection(l1,C)={A,B};PointOnCurve(LeftFocus(C),l);PointOnCurve(UpperVertex(C),l);IsDiameter(LineSegmentOf(A,B),G);IsIntersect(l,G)", "coords": {} }, { "index": 1017, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1017.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;a: Number;b: Number;P: Point;H: Point;F1: Point;F2:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,RightPart(C));L:Line;PointOnCurve(P,L);IsPerpendicular(L,OneOf(Asymptote(C)));FootPoint(L,OneOf(Asymptote(C)))=H;Min(Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, H))) = 4*a", "coords": {} }, { "index": 1018, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1019, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1019.png", "answer": "-2", "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/a^2 + y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(Gamma) = sqrt(2)/2;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), Gamma) = True;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;D: Point;E: Point;F: Point;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, F)) = 1", "coords": {} }, { "index": 1020, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1021, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1021.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(C)) = {A,B};IsPerpendicular(LineSegmentOf(A,F1),LineSegmentOf(A,B));4*Abs(LineSegmentOf(A,F1))=3*Abs(LineSegmentOf(A,B))", "coords": {} }, { "index": 1022, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1022.png", "answer": "1/2", "fact_expr": "M: Ellipse;b: Number;a: Number;c: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;PointOnCurve(P,M);Max(DotProduct(VectorOf(P, F1), VectorOf(P,F2))) = 3*c^2;c^2 = a^2-b^2", "coords": {} }, { "index": 1023, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1023.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 16);A: Point;B: Point;P: Point;Coordinate(P) = (2, -1);PointOnCurve(P, LineOf(A,B));Intersection(LineOf(A,B),G)={A,B};Slope(LineOf(A,B)) = 1/2", "coords": { "P": [ 2.0, -1.0 ] } }, { "index": 1024, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1024.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;P: Point;O: Origin;B:Point;A:Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(P) = (x0,y0);PointOnCurve(P,G);WhenMin(Abs(LineSegmentOf(A,B)));x0:Number;y0:Number;l1:Line;TangentOfPoint(P,G)=l1;Expression(l1)=(x0*x/4+y0=1);Intersection(l2,yAxis)=A;Intersection(l1,yAxis)=B", "coords": {} }, { "index": 1025, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1025.png", "answer": "5/2", "fact_expr": "G: Parabola;F: Point;P: Point;y1: Number;Expression(G) = (y^2 = 2*x);Coordinate(P) = (2, y1);Focus(G) = F;PointOnCurve(P, G)", "coords": {} }, { "index": 1026, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1026.png", "answer": "y=pm*sqrt(6)*x", "fact_expr": "F2: Point;F1: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);l: Line;Intersection(l, LeftPart(C)) = P;Intersection(l, RightPart(C)) = Q;Q: Point;P: Point;AngleOf(Q, P, F2) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, Q)) = Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1027, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1027.png", "answer": "x^2/(4*a^2)+y^2/(4*b^2)=1", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;Q: Point;P: Point;F1: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};VectorOf(O, Q) = VectorOf(P, F1) + VectorOf(P, F2)", "coords": {} }, { "index": 1028, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1028.png", "answer": "y^2=-12*x", "fact_expr": "F: Circle;Expression(F) = (y^2 + (x + 3)^2 = 1);l: Line;Expression(l) = (x = 2);M: Circle;IsTangent(l, M);IsOutTangent(F, M);M1: Point;Center(M) = M1", "coords": {} }, { "index": 1029, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1029.png", "answer": "y=pm*(3/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)", "coords": {} }, { "index": 1030, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1030.png", "answer": "sqrt(5)", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;M: Point;P: Point;l1: Line;l2: Line;Expression(C) = (y^2 = 4*x);Expression(G) = (x=-2);Coordinate(M) = (0, 1);PointOnCurve(P,G);TangentOfPoint(P, C) = {l1, l2};TangentPoint(l1,C)=A;TangentPoint(l2,C)=B", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 1031, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "y", "x_coef": 5.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1031.png", "answer": "sqrt(15)/5", "fact_expr": "C: Ellipse;Expression(C) = (x^2/5 + y^2/9 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));r: Number;Radius(InscribedCircle(TriangleOf(P, F1, F2))) = r", "coords": {} }, { "index": 1032, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1032.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G), xAxis);H: Line;Expression(H) = (2*x - y = 0);A: Point;B: Point;Intersection(H, G) = {A, B};P: Point;Coordinate(P) = (1, 2);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 2.0 ] } }, { "index": 1033, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1033.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;LeftVertex(C) = A;O: Origin;G: Circle;IsDiameter(LineSegmentOf(O,F),G);Intersection(G,OneOf(Asymptote(C))) = P;P: Point;e: Number;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, F));Eccentricity(C) = e", "coords": {} }, { "index": 1034, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1034.png", "answer": "y=pm*2*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(C))=F;OneOf(Endpoint(ImageinaryAxis(C)))=B;PointOnCurve(MidPoint(LineSegmentOf(F,B)), C)", "coords": {} }, { "index": 1035, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1035.png", "answer": "-8", "fact_expr": "H: Parabola;Expression(H) = (x^2 = a*y);a: Number;G: Hyperbola;Expression(G) = (-x^2 + y^2/3 = 1);Focus(H) = LowerFocus(G)", "coords": {} }, { "index": 1036, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1037, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1038, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1038.png", "answer": "x^2/4-(3*y^2)/4=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*sqrt(3)/3*x);Distance(Vertex(G), Asymptote(G)) = 1", "coords": {} }, { "index": 1039, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1040, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1040.png", "answer": "pm*3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);A: Point;RightVertex(C) = A;PointOnCurve(A, l);l: Line;Intersection(l, Asymptote(C)) = {P, Q};P: Point;Q: Point;VectorOf(P, A) = 2*VectorOf(A, Q)", "coords": {} }, { "index": 1041, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1041.png", "answer": "sqrt(13)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);AngleOf(F1,P,F2)=2*pi/3;Abs(LineSegmentOf(F1, P)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1042, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1042.png", "answer": "-1", "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/5 + y^2 = 1);F1: Point;F2: Point;LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;P: Point;PointOnCurve(P, Gamma);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 3", "coords": {} }, { "index": 1043, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1043.png", "answer": "6 +sqrt(10)", "fact_expr": "C: Ellipse;M: Point;N: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;UpperVertex(C)=N;Abs(VectorOf(M, N))^2 = 2*DotProduct(VectorOf(M, F1), VectorOf(M, F2))", "coords": {} }, { "index": 1044, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1044.png", "answer": "{25, -7}", "fact_expr": "C: Curve;a: Number;Expression(C) = (y^2/9 + x^2/a = 1);FocalLength(C) = 8", "coords": {} }, { "index": 1045, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 9.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1045.png", "answer": "[7,9)", "fact_expr": "H: Curve;Expression(H) = (x = sqrt(a - y^2));a: Number;G: Ellipse;Expression(G) = (x^2/9 + y^2/7 = 1);NumIntersection(H, G) = 2", "coords": {} }, { "index": 1046, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.708203932499369, "b": 4.47213595499958, "major_axis": "x", "x_coef": 45.0, "y_coef": 20.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1046.png", "answer": "(-3,4)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/45 + y^2/20 = 1);P:Point;PointOnCurve(P,G);F1:Point;F2:Point;Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Quadrant(P)=2", "coords": {} }, { "index": 1047, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1047.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G)=(x^2/m+y^2/9=1);m:Number;m>9;C:Hyperbola;Expression(C)=(x^2/n-y^2/9=1);n>0;n:Number;F1:Point;F2:Point;P:Point;Focus(G)={F1,F2};Focus(C)={F1,F2};OneOf(Intersection(G,C))=P", "coords": {} }, { "index": 1048, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1048.png", "answer": "8/3", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, l) = A;OneOf(Intersection(G,C))=B;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 1049, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1049.png", "answer": "sqrt(3)-1", "fact_expr": "l: Line;C: Ellipse;a: Number;b: Number;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Expression(l)=(y = sqrt(3)*x);Intersection(l, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 1050, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1051, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1051.png", "answer": "x^2/24+y^2/8=1", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;M: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (4, 0);Coordinate(M) = (1, -1);RightFocus(E) = F;PointOnCurve(F, G);Intersection(G, E) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "F": [ 4.0, 0.0 ], "M": [ 1.0, -1.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1052, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1052.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 1053, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1053.png", "answer": "y^2=8*x", "fact_expr": "C: Parabola;P: Point;O: Origin;m:Number;Coordinate(P) = (4, m);Vertex(C) = O;PointOnCurve(Focus(C), xAxis);PointOnCurve(P, C);Distance(P, Focus(C)) = 6", "coords": {} }, { "index": 1054, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1055, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1055.png", "answer": "10", "fact_expr": "G: Parabola;Focus(G) = F;Coordinate(F) = (0, 1);F: Point;P: Point;Coordinate(P) = (0, 2);l: Line;PointOnCurve(P,l) = True;Intersection(l, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 4*sqrt(6)", "coords": { "F": [ 0.0, 1.0 ], "P": [ 0.0, 2.0 ] } }, { "index": 1056, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1056.png", "answer": "[1,9)", "fact_expr": "G: Ellipse;m: Real;H: Line;k: Number;Expression(G) = (x^2/9 + y^2/m = 1);Expression(H) = (y = k*x + 1);PointOnCurve(Focus(G),xAxis);IsIntersect(H,G)", "coords": {} }, { "index": 1057, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1057.png", "answer": "14", "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/36 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;Distance(P, F1) = 6;OneOf(Focus(G)) = F2;Negation(F1= F2)", "coords": {} }, { "index": 1058, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1058.png", "answer": "1/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;c:Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G)={F1,F2};P: Point;PointOnCurve(P, G);Area(TriangleOf(P, F1, F2)) =(sqrt(2)/2)*b^2", "coords": {} }, { "index": 1059, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1059.png", "answer": "5/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(C) = F;B1: Point;B2: Point;ImageinaryAxis(C) = LineSegmentOf(B1, B2);M: Point;O: Origin;MidPoint(LineSegmentOf(O, B1)) = M;G: Line;PointOnCurve(F, G);PointOnCurve(M, G);A: Point;Intersection(G, C) = A;LineSegmentOf(F, M) = 2*LineSegmentOf(M, A)", "coords": {} }, { "index": 1060, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1060.png", "answer": "x^2-y^2/2=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Slope(OneOf(Asymptote(G))) = sqrt(2);H: Parabola;Expression(H) = (y^2 = 4*(sqrt(3)*x));RightFocus(G) = Focus(H)", "coords": {} }, { "index": 1061, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1062, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1063, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1063.png", "answer": "(-3, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x)", "coords": {} }, { "index": 1064, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1064.png", "answer": "sqrt(10)/2", "fact_expr": "H: Line;Expression(H) = (y = 1 - x);G: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(H, G) = {A, B};A: Point;B: Point;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True;O: Origin;e: Number;Eccentricity(G) = e;In(e, [1/2, sqrt(3)/2])", "coords": {} }, { "index": 1065, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_1065.png", "answer": "y = pm*2*x", "fact_expr": "G: Hyperbola;a: Number;H: Point;a>0;Expression(G) = (-y^2/16 + x^2/a^2 = 1);Coordinate(H) = (2, 0);PointOnCurve(H, G)", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 1066, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1066.png", "answer": "5", "fact_expr": "G: Parabola;H: Line;a: Real;O: Origin;A: Point;B: Point;F:Point;Expression(H) = (x = a);Expression(G) = (y^2 = 4*x);Intersection(H, G) = {A, B};Focus(G)=F;Orthocenter(TriangleOf(O,A,B))=F", "coords": {} }, { "index": 1067, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1067.png", "answer": "sqrt(2)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;A: Point;Coordinate(A) = (-4, 0);P: Point;PointOnCurve(P, C)", "coords": { "A": [ -4.0, 0.0 ] } }, { "index": 1068, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1068.png", "answer": "3", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y = x^2/4);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = 4", "coords": {} }, { "index": 1069, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1070, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1070.png", "answer": "1", "fact_expr": "A: Point;B: Point;C: Curve;Expression(C) = (y = x^2/4);PointOnCurve(A, C);PointOnCurve(B, C);XCoordinate(A) + XCoordinate(B) = 4", "coords": {} }, { "index": 1071, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1071.png", "answer": "{1,13}", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/7 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C) = True;P: Point;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 1072, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1072.png", "answer": "3", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C)=(y^2 = 2*(p*x));p:Number;p>0;Focus(C) = F;PointOnCurve(F, l);Inclination(l) = ApplyUnit(60, degree);Intersection(l, C) = {A, B};Quadrant(A)=1;Quadrant(B)=4", "coords": {} }, { "index": 1073, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1073.png", "answer": "7/2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 1074, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1075, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1075.png", "answer": "8", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1,l) = True;Intersection(l,G) = {A,B};Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 1076, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 4.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1076.png", "answer": "12", "fact_expr": "C: Ellipse;Expression(C) = (x^2/36 + y^2/16 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C)", "coords": {} }, { "index": 1077, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1077.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;F1: Point;A: Point;B: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(G, L1) = A;Intersection(G,L2) =B;VectorOf(F1, A) = VectorOf(A, B);DotProduct(VectorOf(F1,B),VectorOf(F2,B))=0", "coords": {} }, { "index": 1078, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1079, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1079.png", "answer": "13", "fact_expr": "G: Parabola;F: Point;A: Point;P: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;Coordinate(A)=(6,3);PointOnCurve(P,G);Negation(PointOnCurve(P,LineOf(A,F)))", "coords": { "A": [ 6.0, 3.0 ] } }, { "index": 1080, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1080.png", "answer": "3", "fact_expr": "H: Line;G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 1081, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1081.png", "answer": "8+2*sqrt(7)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Negation(P=Endpoint(MajorAxis(G)))", "coords": {} }, { "index": 1082, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1082.png", "answer": "y=1", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4)", "coords": {} }, { "index": 1083, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1083.png", "answer": "x^2/9-y^2/3=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H)=(x^2 + 4*y^2 = 16);Focus(H) = Focus(G);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 1084, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1084.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F: Point;N: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L:Line;L1:Line;L2:Line;PointOnCurve(F, L);IsPerpendicular(L,L1);FootPoint(L,L1)=M;Intersection(L,L2)=N;2*VectorOf(M, F) = VectorOf(F, N);OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2)", "coords": {} }, { "index": 1085, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1085.png", "answer": "3*x+2*y-4=0", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;PointOnCurve(A,C) = True;PointOnCurve(B,C) = True;Negation(A=B);P: Point;Coordinate(P) = (1, 1/2);VectorOf(A, P) = VectorOf(P, B)", "coords": { "P": [ 1.0, 0.5 ] } }, { "index": 1086, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1086.png", "answer": "8", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/24 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, RightPart(C))", "coords": {} }, { "index": 1087, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1087.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1)", "coords": {} }, { "index": 1088, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1088.png", "answer": "x+2*y-3=0", "fact_expr": "P: Point;Coordinate(P) = (1, 1);L: Line;PointOnCurve(P, L) = True;G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);Intersection(L, G) = {M, N};M: Point;N: Point;MidPoint(LineSegmentOf(M, N)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 1089, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1089.png", "answer": "(3+sqrt(10))/4", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F: Point;Q: Point;A: Point;B: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C) = F;PointOnCurve(A,l);Intersection(l,OneOf(Asymptote(C)))=Q;OneOf(Intersection(LineOf(Q,F), C)) = B;DotProduct(VectorOf(A, Q), VectorOf(A, B)) = DotProduct(VectorOf(A, Q), VectorOf(F, B));VectorOf(B, Q) = 4*VectorOf(F, Q);Eccentricity(C) = e", "coords": {} }, { "index": 1090, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5002657771110535, "direction": "right" }, "optimization": { "final_loss": 8.410525879298802e-06, "converged": true, "iterations": 102 }, "output_path": "results/test/parabola/problem_1090.png", "answer": "5/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;Coordinate(M) = (2, 2);PointOnCurve(M, G)", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 1091, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1091.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;H: Line;O: Origin;M: Point;a: Number;Negation(a=0);Coordinate(M) = (a, 0);PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A, B};k1: Number;k2: Number;Slope(LineSegmentOf(O, A)) = k1;Slope(LineSegmentOf(O, B)) = k2;k1*k2 = -2*p", "coords": {} }, { "index": 1092, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1092.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 3", "coords": {} }, { "index": 1093, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1094, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1094.png", "answer": "4*sqrt(2)", "fact_expr": "C: Ellipse;G: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 1095, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1095.png", "answer": "7/2", "fact_expr": "F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 8", "coords": {} }, { "index": 1096, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1096.png", "answer": "4", "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;Expression(C) = (x^2 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1097, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0599746704101562, "b": 2.576634645462036, "orientation": "horizontal" }, "optimization": { "final_loss": 9.342352313979063e-07, "converged": true, "iterations": 116 }, "output_path": "results/test/hyperbola/problem_1097.png", "answer": "4*sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/4 + x^2/m = 1);m: Number;m>0;F1: Point;F2: Point;Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Focus(C) = {F1, F2};M: Point;PointOnCurve(M, C);AngleOf(F1, M, F2) = ApplyUnit(60, degree)", "coords": { "F1": [ -4.0, 0.0 ], "F2": [ 4.0, 0.0 ] } }, { "index": 1098, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1098.png", "answer": "12", "fact_expr": "C: Parabola;B: Point;A: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,LineOf(A,B));PointOnCurve(F,LineOf(D,E));Intersection(LineOf(A,B),C)={A,B};Intersection(LineOf(D,E),C)={D,E};Slope(LineOf(A,B))=k1;Slope(LineOf(D,E))=k2;k1:Number;k2:Number;k1^2+k2^2=4", "coords": {} }, { "index": 1099, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 40.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1099.png", "answer": "y=-40", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 160*y)", "coords": {} }, { "index": 1100, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1101, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1102, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1102.png", "answer": "(6,8)", "fact_expr": "l: Line;C: Ellipse;A: Point;F: Point;B: Point;k:Number;Negation(k=0);Expression(C) = (x^2/4 + y^2 = 1);LeftFocus(C) = F;Expression(l)=(y = k*x) ;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 1103, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1103.png", "answer": "7", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (2*x + y - 4 = 0);Intersection(G, H) = {A, B};Focus(G) = F", "coords": {} }, { "index": 1104, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1105, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1105.png", "answer": "(1,sqrt(3))", "fact_expr": "A1: Point;A2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > 0;b > 0;LeftVertex(C) = A1;RightVertex(C) = A2;M: Point;PointOnCurve(M, C);k1: Number;k2: Number;Slope(LineOf(M, A1)) = k1;Slope(LineOf(M, A2)) = k2;k1 * k2 < 2", "coords": {} }, { "index": 1106, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1106.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;e: Number;Eccentricity(G) = e;Expression(Asymptote(G)) = (y = pm*x*sqrt(6))", "coords": {} }, { "index": 1107, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1108, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1108.png", "answer": "40", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2 - 16*y^2 = 16);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 12", "coords": {} }, { "index": 1109, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1109.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G))=Focus(H);Distance(Focus(G),Asymptote(G))=1", "coords": {} }, { "index": 1110, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1111, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1112, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1112.png", "answer": "10", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(M, G);XCoordinate(M) = 6;F: Point;Focus(G) = F", "coords": {} }, { "index": 1113, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1113.png", "answer": "13", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (y^2 + (x + 4)^2 = 4);K: Circle;Expression(K) = (y^2 + (x - 4)^2 = 1);Z1: Line;Z2: Line;TangentOfPoint(P, H) = Z1;TangentOfPoint(P, K) = Z2;M: Point;TangentPoint(Z1, H) = M;N: Point;TangentPoint(Z2, K) = N", "coords": {} }, { "index": 1114, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1114.png", "answer": "sqrt(7)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F2: Point;B: Point;O: Origin;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);Intersection(l, C) = {A, B};IsPerpendicular(LineSegmentOf(B, F1),LineSegmentOf(O, A));Abs(LineSegmentOf(A, F2)) = Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 1115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1115.png", "answer": "4*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {P, Q};P: Point;Q: Point;Projection(P, Directrix(C)) = M;Projection(Q, Directrix(C)) = N;M: Point;N: Point", "coords": {} }, { "index": 1116, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1116.png", "answer": "27", "fact_expr": "G: Hyperbola;m: Real;Negation(m=0);Expression(G) = (x^2/9 - y^2/m = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 1117, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1117.png", "answer": "3*sqrt(3)", "fact_expr": "G: Parabola;M: Point;x1: Number;y1: Number;Expression(G) = (y^2 = 6*x);Coordinate(M) = (x1, y1);PointOnCurve(M, G);Distance(M, Focus(G)) = 9/2;O: Origin", "coords": {} }, { "index": 1118, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1119, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1120, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1120.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)", "coords": {} }, { "index": 1121, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1121.png", "answer": "1/2", "fact_expr": "l: Line;G: Ellipse;a: Number;b: Number;A: Point;F: Point;P: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;RightFocus(G) = F;PointOnCurve(F, l);IsPerpendicular(xAxis, l);PointOnCurve(P, l);AngleOf(A, P, O) = ApplyUnit(30, degree)", "coords": {} }, { "index": 1122, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.5, "b": 0.8660254037844386, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1122.png", "answer": "", "fact_expr": "l: Line;G: Hyperbola;H: Ellipse;Expression(G) = (12*x^2 - 4*y^2 = 3);Expression(l) = (y = x + 3);PointOnCurve(P,l);PointOnCurve(P,H);Focus(G)=Focus(H);WhenMin(MajorAxis(H));P:Point", "coords": {} }, { "index": 1123, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1123.png", "answer": "3*x+pm*2*y=0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)", "coords": {} }, { "index": 1124, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1124.png", "answer": "pi", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 1125, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1125.png", "answer": "3", "fact_expr": "G: Parabola;l1: Line;l2: Line;P: Point;Expression(G) = (y = x^2/4);Expression(l1)=(4*x - 3*y - 12 = 0);Expression(l2)=(y+1=0);PointOnCurve(P, G)", "coords": {} }, { "index": 1126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1127, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1127.png", "answer": "[sqrt(2)/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;e:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree);Eccentricity(G)=e", "coords": {} }, { "index": 1128, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1129, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1129.png", "answer": "y=-a/4", "fact_expr": "G: Parabola;a: Number;Expression(G) = (x^2 = a*y);Negation(a=0)", "coords": {} }, { "index": 1130, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1130.png", "answer": "(1, 1)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2);Expression(H) = (2*x - y = 4);Z: Point;PointOnCurve(Z, G);WhenMin(Distance(Z, H))", "coords": {} }, { "index": 1131, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1132, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1132.png", "answer": "[-sqrt(3)/3,sqrt(3)/3]", "fact_expr": "G: Hyperbola;H: Line;F: Point;Expression(G) = (x^2/12 - y^2/4 = 1);RightFocus(G) = F;PointOnCurve(F,H);NumIntersection(H,RightPart(G))=1", "coords": {} }, { "index": 1133, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1133.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Real;PointOnCurve(Focus(G), yAxis);Eccentricity(G) = sqrt(3)/2", "coords": {} }, { "index": 1134, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1134.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 3", "coords": {} }, { "index": 1135, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1135.png", "answer": "sqrt(10)/4", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(E) = F;l: Line;O: Origin;PointOnCurve(O, l);P: Point;Q: Point;Intersection(l, E) = {P, Q};Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F));AngleOf(P, F, Q) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1136, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1136.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(Focus(G),Asymptote(G)) = a", "coords": {} }, { "index": 1137, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1137.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/5 = 1)", "coords": {} }, { "index": 1138, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1138.png", "answer": "(2,0)", "fact_expr": "C: Ellipse;P: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 1", "coords": {} }, { "index": 1139, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1139.png", "answer": "[1,3]", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F;PointOnCurve(P, G)", "coords": {} }, { "index": 1140, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1140.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;p: Number;O: Origin;M: Point;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(P, G);PointOnCurve(M, LineSegmentOf(P,F));Abs(LineSegmentOf(P, M)) = 2*Abs(LineSegmentOf(M, F))", "coords": {} }, { "index": 1141, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1142, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1142.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;l: Line;A: Point;B: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(H), l);IsPerpendicular(l,xAxis);Intersection(l, H) = {A, B};LineSegmentOf(A, B) = a/2", "coords": {} }, { "index": 1143, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1143.png", "answer": "4", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/25 + y^2/m^2 = 1);FocalLength(G) = 6;m>0;m<5", "coords": {} }, { "index": 1144, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1144.png", "answer": "1", "fact_expr": "G: Hyperbola;n: Number;H: Ellipse;m: Number;F1: Point;P: Point;F2: Point;n>0;Expression(G) = (-y^2 + x^2/n = 1);m>1;Expression(H) = (y^2 + x^2/m = 1);Focus(G) = {F1,F2};Focus(H) = {F1,F2};OneOf(Intersection(G,H))=P", "coords": {} }, { "index": 1145, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1145.png", "answer": "13", "fact_expr": "G: Hyperbola;H: Ellipse;m: Number;n: Number;P: Point;F1: Point;F2: Point;m>0;n>0;Expression(H) = (y^2/9 + x^2/m^2 = 1);Expression(G) = (-y^2/4 + x^2/n^2 = 1);Focus(H) = {F1, F2};Focus(G) = {F1, F2};OneOf(Intersection(H,G))=P", "coords": {} }, { "index": 1146, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1146.png", "answer": "3", "fact_expr": "G: Hyperbola;A: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A,LeftPart(G));IsPerpendicular(LineSegmentOf(A,F1),LineSegmentOf(A,F2))", "coords": {} }, { "index": 1147, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1147.png", "answer": "7*x+8*y-22=0", "fact_expr": "l: Line;G: Ellipse;a: Number;b: Number;A: Point;B: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (2, 1);FocalLength(G)=6;MinorAxis(G)=(sqrt(7)/4)*MajorAxis(G);Intersection(l,G)={A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 1148, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1148.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;L: Line;PointOnCurve(P, L) = True;IsPerpendicular(L, Directrix(G)) = True;FootPoint(L, Directrix(G)) = M;M: Point;Abs(LineSegmentOf(P, M)) = 5;F: Point;Focus(G) = F", "coords": {} }, { "index": 1149, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 9.0, "b": 5.196152422706632, "major_axis": "x", "x_coef": 81.0, "y_coef": 27.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1149.png", "answer": "36", "fact_expr": "G: Ellipse;H:Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/81 + y^2/27 = 1);Focus(G) = {F1, F2};PointOnCurve(F1,H);Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 1150, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1150.png", "answer": "x^2/16+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)/2;Length(MinorAxis(G))=4", "coords": {} }, { "index": 1151, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1151.png", "answer": "(5,pm*2*sqrt(5))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P:Point;Distance(P,Focus(G))=6;PointOnCurve(P,G)", "coords": {} }, { "index": 1152, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1152.png", "answer": "(2*sqrt(3))/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(C) = F;A: Point;Coordinate(A) = (0, 2*b);Inclination(LineOf(A, F)) = ApplyUnit(45, degree)", "coords": {} }, { "index": 1153, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1153.png", "answer": "sqrt(5)", "fact_expr": "C: Parabola;A: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (0, 2);PointOnCurve(P, C)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 1154, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1155, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1155.png", "answer": "x^2/20-y^2/5=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 1);FocalLength(C) = 10;PointOnCurve(P, Asymptote(C))", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 1156, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1157, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1157.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;m>0;Expression(G) = (x^2/4 - y^2/m^2 = 1);Expression(H) = (y^2 = 16*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 1158, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1159, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1160, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1160.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H) = True;Intersection(H, G) = {A, B};A: Point;B: Point;l: Line;Directrix(G) = l;Intersection(H, l) = C;C: Point;Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 1161, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1161.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,RightPart(G));DotProduct((VectorOf(O, F2) + VectorOf(O, P)),VectorOf(F2, P)) = 0;Abs(VectorOf(P, F1)) = sqrt(3)*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 1162, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1162.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;a: Number;O: Origin;F: Point;P: Point;Expression(C) = (y^2/3 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(P,C);Abs(LineSegmentOf(O,F))=Abs(LineSegmentOf(F,P))", "coords": {} }, { "index": 1163, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1163.png", "answer": "28/3", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Expression(l) = (y = sqrt(3)*x + b);Focus(C) = F;Distance(F, l) = sqrt(3);Intersection(l, C) = {A, B};b:Number", "coords": {} }, { "index": 1164, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1164.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 8*y);F: Point;Focus(C) = F;O: Origin;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F), LineSegmentOf(O, F))", "coords": {} }, { "index": 1165, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1165.png", "answer": "10", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 1166, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1166.png", "answer": "32", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/8 = 1);F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, LeftPart(C));A: Point;Coordinate(A) = (0, 6*sqrt(6));PointOnCurve(A, yAxis)", "coords": { "A": [ 0.0, 14.696938456699067 ] } }, { "index": 1167, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1167.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;I: Point;Incenter(TriangleOf(P,F1,F2)) = I;lambda: Real;Area(TriangleOf(P,F1,F2)) = 2*Area(TriangleOf(I,P,F2)) + (1+1/lambda)*Area(TriangleOf(I,F1,F2))", "coords": {} }, { "index": 1168, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/test/hyperbola/problem_1168.png", "answer": "{sqrt(6)/2,sqrt(3)}", "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y=sqrt(2)*x)", "coords": {} }, { "index": 1169, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1169.png", "answer": "3*sqrt(6)/8", "fact_expr": "l: Line;PointOnCurve(Focus(G), l) = True;Intersection(l,G) = {P,Q};G: Ellipse;Expression(G) = (x^2 + y^2/2 = 1);P: Point;Q: Point;M: Point;Intersection(PerpendicularBisector(LineSegmentOf(P,Q)),xAxis) = M", "coords": {} }, { "index": 1170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1170.png", "answer": "y=pm*(1/2)*x\n12", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 - y^2/4 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1171, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1172, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1172.png", "answer": "[sqrt(2)/2,1]", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;Intersection(Directrix(G),yAxis) = M;M: Point;PointOnCurve(N,G) = True;N: Point;lambda: Number;Abs(LineSegmentOf(N, F)) = lambda*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 1173, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1173.png", "answer": "1/4", "fact_expr": "G: Ellipse;a: Number;M: Point;A: Point;B: Point;N: Point;F1: Point;F2: Point;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);RightVertex(G) = A;LowerVertex(G) = B;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, G);PointOnCurve(N, G);Max(Area(TriangleOf(M, A, B))) = 1 + sqrt(2)", "coords": {} }, { "index": 1174, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1174.png", "answer": "sqrt(5)", "fact_expr": "C2: Hyperbola;C1: Parabola;A: Point;p: Number;a: Number;b: Number;a > 0;b > 0;p > 0;Expression(C1) = (y^2 = 2*p*x);Expression(C2) = (x^2/a^2 - y^2/b^2 = 1);Intersection(C1, OneOf(Asymptote(C2))) = A;Distance(A, Directrix(C1)) = p", "coords": {} }, { "index": 1175, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1175.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;a > 0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0)", "coords": {} }, { "index": 1176, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1176.png", "answer": "1", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;Coordinate(M) = (-1, 2);PointOnCurve(Focus(C), G);Slope(G) = k;k: Number;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 1177, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1178, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1178.png", "answer": "[2*a, (4*sqrt(3)*a)/3]", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;e: Number;Eccentricity(C) = e;e = 2;G: Line;PointOnCurve(F2, G);A: Point;B: Point;Intersection(G, RightPart(C)) = {A, B};Quadrant(A) = 1;M: Point;N: Point;Incenter(TriangleOf(A, F1, F2)) = M;Incenter(TriangleOf(B, F1, F2)) = N;a>0;b>0", "coords": {} }, { "index": 1179, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.24967060983181, "direction": "right" }, "optimization": { "final_loss": 6.555423624376999e-06, "converged": true, "iterations": 116 }, "output_path": "results/test/parabola/problem_1179.png", "answer": "1/2", "fact_expr": "C: Parabola;p: Number;G: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(G) = (1, 1);PointOnCurve(G, C)", "coords": { "G": [ 1.0, 1.0 ] } }, { "index": 1180, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1180.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;P: Point;Q: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {P, Q};IsPerpendicular(VectorOf(P,F2),VectorOf(P,Q));Perimeter(TriangleOf(P,Q,F2))=12*a", "coords": {} }, { "index": 1181, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1182, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_1182.png", "answer": "x^2/36-y^2/64=1", "fact_expr": "G: Hyperbola;F: Point;O:Origin;Center(G)=O;Coordinate(F) = (10, 0);OneOf(Focus(G))=F;Expression(Asymptote(G)) = (y = pm*(4/3)*x)", "coords": { "F": [ 10.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1183, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1183.png", "answer": "2", "fact_expr": "Q: Point;Coordinate(Q) = (-2*sqrt(2), 0);G: Parabola;Expression(G) = (x^2 = -4*y);P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;PointOnCurve(P, G) = True", "coords": { "Q": [ -2.8284271247461903, 0.0 ] } }, { "index": 1184, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1184.png", "answer": "3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Parabola;Expression(G) = (x^2 = 4*y);Distance(Focus(G), Asymptote(C)) = 1/3", "coords": {} }, { "index": 1185, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1185.png", "answer": "2*sqrt(3)+3*sqrt(2)", "fact_expr": "P: Point;m: Number;Coordinate(P) = (m, 2);C: Hyperbola;Expression(C) = (x^2/2 - y^2/2 = 1);PointOnCurve(P, RightPart(C));F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Q: Point;Intersection(LineSegmentOf(P, F1), LeftPart(C)) = Q", "coords": {} }, { "index": 1186, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1186.png", "answer": "sqrt(10)/ 2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, G) = True;AngleOf(F1, A, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(A, F2))", "coords": {} }, { "index": 1187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1187.png", "answer": "4", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Line;P: Point;Q: Point;F2: Point;A: Point;B: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1,H);IsPerpendicular(xAxis, H);Intersection(H, LeftPart(G)) = {A, B};Intersection(LineSegmentOf(A, F2), yAxis) = P ;Intersection(LineSegmentOf(B, F2), yAxis) = Q;Perimeter(TriangleOf(P, Q, F2)) = 16", "coords": {} }, { "index": 1188, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1188.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(Asymptote(G)) = ((2*y)+(3*(pm*x)) = 0)", "coords": {} }, { "index": 1189, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.928203230275509, "major_axis": "x", "x_coef": 100.0, "y_coef": 48.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1189.png", "answer": "48", "fact_expr": "C: Ellipse;Q: Point;F1: Point;F2: Point;Expression(C) = (x^2/100 + y^2/48 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(Q, C);Abs(LineSegmentOf(Q, F1)) = (2/3)*Abs(LineSegmentOf(Q, F2))", "coords": {} }, { "index": 1190, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1190.png", "answer": "9", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p > 0;F: Point;Focus(C) = F;Distance(F, Directrix(C)) = 4;PointOnCurve(F, l) = True;PointOnCurve(R, l) = True;R: Point;Coordinate(R) = (m, 0);m: Number;l: Line;Intersection(C, l) = {P, Q};P: Point;Q: Point;VectorOf(R, P) = VectorOf(P, F)", "coords": {} }, { "index": 1191, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1192, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1192.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 1193, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_1193.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 1194, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1194.png", "answer": "{sqrt(7)/2, sqrt(21)/3}", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = 1/2;b: Number;a: Number;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)", "coords": {} }, { "index": 1195, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1195.png", "answer": "3+2*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Focus(G) = F;F: Point;PointOnCurve(F,H) = True;H: Line;Intersection(H,G) = {A,B};A: Point;B: Point", "coords": {} }, { "index": 1196, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1197, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1198, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1198.png", "answer": "a/2-p/2", "fact_expr": "G: Parabola;p: Number;a:Number;A: Point;B: Point;M: Point;p>0;a>=2*p;Expression(G) = (y^2 = 2*(p*x));IsChordOf(LineSegmentOf(A,B),G);Length(LineSegmentOf(A,B))=a;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 1199, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1199.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;A: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 0);PointOnCurve(A, Directrix(C))", "coords": { "A": [ -2.0, 0.0 ] } }, { "index": 1200, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1200.png", "answer": "2*sqrt(15)", "fact_expr": "H: Line;Expression(H) = (y = k*x - 2);k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 1201, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1201.png", "answer": "x=8/3", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -12*x);LeftFocus(G) = Focus(H)", "coords": {} }, { "index": 1202, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1202.png", "answer": "{sqrt(6),sqrt(2)}", "fact_expr": "l: Line;C: Circle;A: Point;O: Origin;B: Point;b: Number;Expression(C) = (x^2 + y^2 = 4);Slope(l)=1;b>0;Intercept(l,yAxis)=b;Intersection(l, C) = {A, B};Area(TriangleOf(A, O, B)) = sqrt(3)", "coords": {} }, { "index": 1203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_1203.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(F2, P));Area(TriangleOf(P, F1, F2)) = 8;Eccentricity(C)=sqrt(5)", "coords": {} }, { "index": 1204, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1204.png", "answer": "(1,(2/3)*sqrt(3))", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 1205, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1205.png", "answer": "4", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 12*x);PointOnCurve(P, G) = True;Q: Point;H: Circle;Expression(H) = (x^2 + (y - 4)^2 = 1);PointOnCurve(Q, H) = True;I: Line;Expression(I) = (x = -3)", "coords": {} }, { "index": 1206, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1207, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1207.png", "answer": "3/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Focus(G), LineSegmentOf(A, B));Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 1208, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1209, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1210, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1210.png", "answer": "3", "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2*sqrt(2), 2);PointOnCurve(A,G)", "coords": { "A": [ 2.8284271247461903, 2.0 ] } }, { "index": 1211, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1211.png", "answer": "", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);F: Point;Focus(C) = F;l: Line;Directrix(C) = l", "coords": {} }, { "index": 1212, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1213, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1213.png", "answer": "(y^2 = 80 - 20*x)&(x <= 4)", "fact_expr": "G: Circle;H: Line;Z: Circle;Expression(G) = (2*x + x^2 + y^2 - 15 = 0);Expression(H) = (x = 5);IsTangent(Z, H);IsOutTangent(Z, G)", "coords": {} }, { "index": 1214, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1214.png", "answer": "x^2/4-y^2/3=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Ellipse;Expression(H) = (x^2/16 + y^2/9 = 1);Focus(G) = Focus(H);Eccentricity(G) = Eccentricity(H)*2", "coords": {} }, { "index": 1215, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1216, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1216.png", "answer": "2*x+y-17=0", "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2 + 4*y^2 = 4);Coordinate(P) = (8, 1);MidPoint(H)=P;IsChordOf(H,G)", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 1217, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1217.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 1218, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1218.png", "answer": "-5", "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Curve;PointOnCurve(P, G);Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 1219, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1220, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1220.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1221, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1221.png", "answer": "3", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, sqrt(15));PointOnCurve(P, G);d1:Number;d2:Number;Distance(P, yAxis) = d1;Distance(P, A) = d2", "coords": { "A": [ 2.0, 3.872983346207417 ] } }, { "index": 1222, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1222.png", "answer": "25/3, 3", "fact_expr": "G: Ellipse;m: Number;m>0;e: Number;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = e;e = sqrt(10)/5", "coords": {} }, { "index": 1223, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.535473169108841, "b": 3.535473169108841, "orientation": "horizontal" }, "optimization": { "final_loss": 4.3822798825688845e-07, "converged": true, "iterations": 93 }, "output_path": "results/test/hyperbola/problem_1223.png", "answer": "(x^2/9-y^2/16=1)&(x>0)", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-5, 0);Coordinate(F2) = (5, 0);PointOnCurve(P,H);Distance(P,F1)-Distance(P,F2)=6;P:Point", "coords": { "F1": [ -5.0, 0.0 ], "F2": [ 5.0, 0.0 ] } }, { "index": 1224, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1224.png", "answer": "sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;O: Origin;Center(C) = O;C: Ellipse;OneOf(Focus(C)) = F;Q: Point;OneOf(Focus(C)) = Q;Negation(F = Q);Eccentricity(C) = sqrt(2)/2;P: Point;PointOnCurve(P, C) = True;AngleOf(F, P, Q) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1225, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1225.png", "answer": "2/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G) ;AngleOf(F1, P, F2) = pi/3;R:Number;r:Number;Radius(CircumCircle(TriangleOf(F1,P,F2)))=R;Radius(InscribedCircle(TriangleOf(F1,P,F2)))=r;R=4*r", "coords": {} }, { "index": 1226, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1226.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;l: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F,l);Inclination(l)=ApplyUnit(30,degree);Intersection(l, G) = {A, B};Length(LineSegmentOf(A,B))=8", "coords": {} }, { "index": 1227, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1227.png", "answer": "6", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l,G)={A,B};XCoordinate(MidPoint(LineSegmentOf(A,B))) = 2", "coords": {} }, { "index": 1228, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1228.png", "answer": "(4,2)", "fact_expr": "A: Point;B: Point;G: Parabola;H: Line;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y = 2);Intersection(H, G) = {A,B}", "coords": {} }, { "index": 1229, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1229.png", "answer": "pm*sqrt(7)", "fact_expr": "C: Ellipse;l: Line;m: Real;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Expression(l) = (y = x + m);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, l);Min(Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2))) = 4", "coords": {} }, { "index": 1230, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1230.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (2*x - y + 1 = 0);IsParallel(OneOf(Asymptote(G)), H) = True", "coords": {} }, { "index": 1231, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1231.png", "answer": "8*sqrt(5)/5", "fact_expr": "G: Ellipse;H: Line;t: Number;F: Point;M: Point;N: Point;Expression(G) = (x^2/5 + y^2/4 = 1);Expression(H) = (x = t);LeftFocus(G)=F;Intersection(H, G) = {M, N};WhenMax(Perimeter(TriangleOf(F,M,N)))", "coords": {} }, { "index": 1232, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1232.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2", "coords": {} }, { "index": 1233, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1233.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Focus(G) = {F1, F2};F1: Point;F2: Point;A: Point;PointOnCurve(A, G)", "coords": {} }, { "index": 1234, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1234.png", "answer": "(sqrt(5),+oo)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x);IsIntersect(G, H)", "coords": {} }, { "index": 1235, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1236, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1236.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 1237, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1237.png", "answer": "5*sqrt(2)+5", "fact_expr": "G: Hyperbola;H: Circle;M: Point;F1: Point;F2:Point;A: Point;Expression(H) = (x^2 + (y - 5)^2 = 1);Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(A, H);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(M, RightPart(G))", "coords": {} }, { "index": 1238, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1238.png", "answer": "25/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);F: Point;Focus(G)=F;M: Point;PointOnCurve(M,G);L:Line;PointOnCurve(M,L);IsPerpendicular(Directrix(G),L);N: Point;FootPoint(Directrix(G),L)=N;Abs(LineSegmentOf(N, F)) = 10", "coords": {} }, { "index": 1239, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1239.png", "answer": "pi/2", "fact_expr": "G: Parabola;p: Number;H: Line;A1: Point;F: Point;B1: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Projection(A,Directrix(G))=A1;Projection(B,Directrix(G))=B1", "coords": {} }, { "index": 1240, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1240.png", "answer": "{1,16}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G) = sqrt(3)/2", "coords": {} }, { "index": 1241, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1241.png", "answer": "(0,4)", "fact_expr": "l: Line;C: Ellipse;O: Origin;M: Point;N: Point;Q: Point;lambda:Real;Expression(C) = (x^2/8 + y^2/4 = 1);Expression(l) = (y = 2*m + x/m);m:Number;Negation(m=0);Intersection(l, C) = {M, N};PointOnCurve(Q, C);VectorOf(O, M) + VectorOf(O, N) = lambda*VectorOf(O, Q);Negation(lambda=0)", "coords": {} }, { "index": 1242, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1242.png", "answer": "pm*4/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 1243, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1243.png", "answer": "-4", "fact_expr": "G: Hyperbola;m: Real;F1: Point;Expression(G) = (y^2/12 + x^2/m = 1);Coordinate(F1) = (0, 4);OneOf(Focus(G)) = F1", "coords": { "F1": [ 0.0, 4.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1244, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1244.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, LeftPart(G));Abs(LineSegmentOf(M, F1)) = 2*Abs(LineSegmentOf(F1, F2));Cos(AngleOf(M, F1, F2)) = -5/16;Eccentricity(G) = e", "coords": {} }, { "index": 1245, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1245.png", "answer": "sqrt(10)", "fact_expr": "G: Ellipse;l: Line;M: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(M) = (4, 2);MidPoint(InterceptChord(l, G)) = M", "coords": { "M": [ 4.0, 2.0 ] } }, { "index": 1246, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1246.png", "answer": "sqrt(2)/2", "fact_expr": "H: Line;Expression(H) = (x - 2*y + 3 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(H, G) = {A, B};A: Point;B: Point;P: Point;Coordinate(P) = (-1, 1);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ -1.0, 1.0 ] } }, { "index": 1247, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1247.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;P: Point;M: Point;F: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);PointOnCurve(P,LineSegmentOf(P,M));IsPerpendicular(LineSegmentOf(P,M),l);FootPoint(LineSegmentOf(P,M),l)=M;Abs(LineSegmentOf(M, P)) = Abs(LineSegmentOf(M, F));Area(TriangleOf(P, M, F)) = sqrt(3)", "coords": {} }, { "index": 1248, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1248.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C) = True;PointOnCurve(A, Z) = True;l: Line;Directrix(C) = l;IsPerpendicular(Z, l) = True;Z: Line;FootPoint(Z, l) = B;B: Point;Intersection(LineSegmentOf(F, B), xAxis) = D;D: Point;Abs(LineSegmentOf(A, F)) = 5", "coords": {} }, { "index": 1249, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1249.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(C))) = ApplyUnit(120, degree)", "coords": {} }, { "index": 1250, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1250.png", "answer": "6", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;M: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = E;PointOnCurve(F, G);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M;Abs(LineSegmentOf(M, E)) = sqrt(11)", "coords": {} }, { "index": 1251, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1251.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;O: Origin;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Tan(AngleOf(F1, P, F2)) = 4*sqrt(3)", "coords": {} }, { "index": 1252, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1252.png", "answer": "9", "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C)", "coords": {} }, { "index": 1253, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1253.png", "answer": "x^2+2*y^2=1", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/2 = 1);P: Point;L: Line;PointOnCurve(P,G);PointOnCurve(P, L);IsPerpendicular(L, xAxis);Q: Point;FootPoint(L, xAxis) = Q;M: Point;MidPoint(LineSegmentOf(P, Q)) = M", "coords": {} }, { "index": 1254, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1254.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Number;F: Point;Expression(G) = (-x^2/9 + y^2/m = 1);Coordinate(F) = (0, 5);OneOf(Focus(G)) = F", "coords": { "F": [ 0.0, 5.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1255, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1255.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Ellipse;H: Line;k: Number;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = k*x + 1)", "coords": {} }, { "index": 1256, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1257, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1257.png", "answer": "x=5*p/2", "fact_expr": "G: Parabola;p: Number;B: Point;A: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*p*x);PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, B));Orthocenter(TriangleOf(A,O,B))=Focus(G)", "coords": {} }, { "index": 1258, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1258.png", "answer": "(2,pm*2*sqrt(2))", "fact_expr": "C: Parabola;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 1259, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1259.png", "answer": "2*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F));G: Circle;IsDiameter(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 1260, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1260.png", "answer": "2*sqrt(2)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1)", "coords": {} }, { "index": 1261, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1261.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b + x^2/a = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(P, F1, F2) = pi/6;AngleOf(P, F2, F1) = pi/3;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 1262, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1262.png", "answer": "1/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-a*y + b*x = 0);Eccentricity(G) = 2;RightVertex(G) = A;RightFocus(G) = F;d1: Number;d2: Number;Distance(A, H) = d1;Distance(F, H) = d2", "coords": {} }, { "index": 1263, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1263.png", "answer": "125/16", "fact_expr": "k: Number;Slope(l) = k;k = 3/4;l: Line;G: Parabola;Expression(G) = (x^2 = 4*y);PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;Intersection(TangentOfPoint(A, G), TangentOfPoint(B, G)) = M", "coords": {} }, { "index": 1264, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1264.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;XCoordinate(H) = 6;PointOnCurve(H, G);Distance(H, Focus(G)) = 8", "coords": {} }, { "index": 1265, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1265.png", "answer": "y=pm*(sqrt(5)/2)*x", "fact_expr": "E: Ellipse;Expression(E) = (x^2/11 + y^2/2 = 1);C: Hyperbola;Expression(C) = (-y^2/5 + x^2/a^2 = 1);a: Number;a>0;Focus(E) = Focus(C)", "coords": {} }, { "index": 1266, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1266.png", "answer": "", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1)", "coords": {} }, { "index": 1267, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1267.png", "answer": "(0, -1/2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*y)", "coords": {} }, { "index": 1268, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1268.png", "answer": "2", "fact_expr": "G: Hyperbola;k: Real;Expression(G) = (-y^2/4 + x^2/k = 1);Coordinate(OneOf(Focus(G)))=(sqrt(6),0)", "coords": {} }, { "index": 1269, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1270, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1270.png", "answer": "pm*4", "fact_expr": "G: Parabola;a: Number;H: Ellipse;Expression(G) = (y^2 = a*x);Expression(H) = (x^2/4 + y^2/3 = 1);OneOf(Focus(H)) = Focus(G)", "coords": {} }, { "index": 1271, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1271.png", "answer": "2", "fact_expr": "l: Line;Slope(l) = 1;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;B: Point;D: Point;Intersection(l, C) = {B, D};M: Point;Coordinate(M)=(1,3);MidPoint(LineSegmentOf(B, D)) = M", "coords": { "M": [ 1.0, 3.0 ] } }, { "index": 1272, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1272.png", "answer": "1/3", "fact_expr": "Gamma: Parabola;l: Line;O: Origin;F: Point;A: Point;B: Point;Expression(Gamma) = (y^2 = 4*x);Expression(l) = (sqrt(3)*x - y - sqrt(3) = 0);Intersection(l,Gamma)={A,B};Intersection(l,xAxis)=F;lambda:Number;mu:Number;lambda<=mu;VectorOf(O, F) = lambda*VectorOf(O, A) + mu*VectorOf(O, B)", "coords": {} }, { "index": 1273, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1274, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1274.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;G: Circle;a:Number;b: Number;H: Line;P: Point;F1: Point;F2: Point;A: Point;Q: Point;Expression(C)=(y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, H);TangentPoint(H,G)=A;Intersection(H,C) = {P,Q};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));a>b;b>0", "coords": {} }, { "index": 1275, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1275.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = 5;O: Origin", "coords": {} }, { "index": 1276, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1276.png", "answer": "12", "fact_expr": "G: Ellipse;l1: Line;l2:Line;A: Point;F: Point;B: Point;C: Point;D: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Expression(l1) = (x - y - 2 = 0);Expression(l2)=(x-y+2=0);LeftFocus(G) = F;Intersection(l1,G)={A,B};Intersection(l2,G)={C,D}", "coords": {} }, { "index": 1277, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1278, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.03125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1278.png", "answer": "(0, 1/32)", "fact_expr": "G: Parabola;Expression(G) = (y = 8*x^2)", "coords": {} }, { "index": 1279, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1279.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;M: Point;PointOnCurve(M, Directrix(C));YCoordinate(M) =3*sqrt(3);N: Point;OneOf(Intersection(LineOf(M, F), C)) = N;VectorOf(M, N) = 2*VectorOf(N, F)", "coords": {} }, { "index": 1280, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1280.png", "answer": "1/2", "fact_expr": "H: Line;Expression(H) = (k*x - y + 1 = 0);k: Number;G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Intersection(H, G) = {A, B};A: Point;B: Point;Negation(A = B);M: Point;Coordinate(M) = (3, 0);PointOnCurve(M, xAxis);Distance(M, A) = Distance(M, B)", "coords": { "M": [ 3.0, 0.0 ] } }, { "index": 1281, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1281.png", "answer": "sqrt(17)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);x1: Number;y1: Number;P: Point;Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": {} }, { "index": 1282, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1283, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1283.png", "answer": "pi/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;Coordinate(M) = (x1, y1);x1: Number;y1: Number;N: Point;Coordinate(N) = (x2, y2);x2: Number;y2: Number;PointOnCurve(M, C);PointOnCurve(N, C);x1 + x2 + 2 = 2*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 1284, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1284.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;F1: Point;F2: Point;P:Point;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(H,G))=P", "coords": {} }, { "index": 1285, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "vertical" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_1285.png", "answer": "y = pm*x/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Eccentricity(C) = sqrt(5)", "coords": {} }, { "index": 1286, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1286.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Line;D: Point;Coordinate(D) = (2, 0);PointOnCurve(D, H);A: Point;B: Point;Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 10;C: Point;MidPoint(LineSegmentOf(A, B)) = C", "coords": { "D": [ 2.0, 0.0 ] } }, { "index": 1287, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1287.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x - y + 6 = 0)", "coords": {} }, { "index": 1288, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1288.png", "answer": "sqrt(13)+2*sqrt(3)", "fact_expr": "G: Hyperbola;M: Point;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(M) = (4, 3);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))", "coords": { "M": [ 4.0, 3.0 ] } }, { "index": 1289, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1289.png", "answer": "sqrt(7)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;C: Point;Coordinate(C) = (0, 2*b);PointOnCurve(B, PerpendicularBisector(LineSegmentOf(A, C)))", "coords": {} }, { "index": 1290, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1290.png", "answer": "y^2/8-x^2/32=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);Z: Hyperbola;Asymptote(G) = Asymptote(Z);H: Point;Coordinate(H) = (2, 3);PointOnCurve(H, Z)", "coords": { "H": [ 2.0, 3.0 ] } }, { "index": 1291, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1291.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H) ;IsPerpendicular(H, xAxis) ;Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F1, F2)) = (sqrt(3)/2)*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 1292, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1292.png", "answer": "y=pm*sqrt(2)/2*x", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;Expression(H) = (x^2/4 + y^2 = 1);Coordinate(P) = (2, 1);Focus(G) = Focus(H);PointOnCurve(P,G)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 1293, "success": false, "error": "\nF_{1}|=\\lambda|M F_{2}|$ $\\left(\\frac{1}{3} \\leq \\lambda \\leq\n ^\nParseException: Expected end of text, found '$' (at char 23), (line:1, col:24)" }, { "index": 1294, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1294.png", "answer": "5*sqrt(2)/2", "fact_expr": "G: Parabola;H: Line;M: Point;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (1, 0);PointOnCurve(M, H);Intersection(H, G) = {A, B};Focus(G) = F;Abs(LineSegmentOf(A, F)) = 6", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 1295, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1295.png", "answer": "8*sqrt(3)", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)", "coords": {} }, { "index": 1296, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1296.png", "answer": "pm*2*sqrt(2)/3", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;k: Real;Expression(C) = (y^2 = 8*x);Expression(G) = (y = k*(x + 2));Intersection(G, C) = {A, B};Focus(C) = F;Abs(VectorOf(F, A)) = 2*Abs(VectorOf(F, B))", "coords": {} }, { "index": 1297, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1297.png", "answer": "(0,-3/2)", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/6)", "coords": {} }, { "index": 1298, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1299, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1299.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;Expression(G)=(x^2/9+y^2/4=1)", "coords": {} }, { "index": 1300, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1300.png", "answer": "(sqrt(2),+oo)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, RightPart(G));Distance(F2, LineOf(A, F1)) = 2*a", "coords": {} }, { "index": 1301, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1302, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1302.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 1303, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.039365291595459, "b": 2.936960895380947, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 2.390299779739223e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_1303.png", "answer": "y = pm*25/4", "fact_expr": "G: Ellipse;O: Origin;Center(G) = O;Eccentricity(G) = 4/5;Coordinate(OneOf(Focus(G)))=(0,4)", "coords": {} }, { "index": 1304, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1304.png", "answer": "0", "fact_expr": "G: Ellipse;H: Line;E: Point;N: Point;M: Point;P: Point;l:Line;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(Center(G),H);Intersection(H,G)={M,N};Quadrant(M)=1;PointOnCurve(M,l);IsPerpendicular(l,xAxis);FootPoint(l,xAxis)=E;OneOf(Intersection(LineOf(N,E),G))=P", "coords": {} }, { "index": 1305, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1305.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;C: Point;Coordinate(C) = (0, sqrt(2)*b);PointOnCurve(B, PerpendicularBisector(LineSegmentOf(A, C)))", "coords": {} }, { "index": 1306, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1306.png", "answer": "(9/4)*p", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l,Directrix(C)) = M;VectorOf(A, B) + VectorOf(A, M) = 0", "coords": {} }, { "index": 1307, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1307.png", "answer": "pm*1", "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;P: Point;F: Point;Q: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = Q;Abs(LineSegmentOf(F, Q)) = 2", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 1308, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1308.png", "answer": "{pi/3,2*pi/3}", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;theta: Number;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};-1/Abs(LineSegmentOf(B, F)) + 1/Abs(LineSegmentOf(A, F)) = 1;Inclination(l) = theta", "coords": {} }, { "index": 1309, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1309.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;p: Number;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (-1, 0);Intersection(Directrix(G), xAxis) = A;Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 1310, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1311, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1312, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1313, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1313.png", "answer": "(1,5/3]", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));IsIntersect(LineOf(P,F1), H)", "coords": {} }, { "index": 1314, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1314.png", "answer": "sqrt(15)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;P: Point;B: Point;O:Origin;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A=B);Negation(A=P);Negation(B=P);PointOnCurve(O,LineSegmentOf(A,B));Slope(LineOf(P, B)) = k2;Slope(LineOf(P,A))=k1;k1:Number;k2:Number;k1*k2=2/3;Eccentricity(G) = e", "coords": {} }, { "index": 1315, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1315.png", "answer": "100/3", "fact_expr": "G: Parabola;H: Ellipse;A: Point;B: Point;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(G)=Center(H);LeftDirectrix(H)=Directrix(G);Intersection(RightDirectrix(H),G)={A,B}", "coords": {} }, { "index": 1316, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1316.png", "answer": "y^2/20+x^2/4=1", "fact_expr": "G: Ellipse;H: Point;Expression(G) = (x^2/9 + y^2/25 = 1);Coordinate(H) = (sqrt(3), sqrt(5));C:Ellipse;Focus(C)=Focus(G);PointOnCurve(H,C)", "coords": { "H": [ 1.7320508075688772, 2.23606797749979 ] } }, { "index": 1317, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1317.png", "answer": "1", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(P, F1)) = M;Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1318, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.1209959984138265, "b": 2.1209959984138265, "orientation": "horizontal" }, "optimization": { "final_loss": 8.099017116797696e-06, "converged": true, "iterations": 98 }, "output_path": "results/test/hyperbola/problem_1318.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G) = {F1, F2};Length(ImageinaryAxis(G)) = sqrt(2)*Length(RealAxis(G))", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 1319, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1319.png", "answer": "pm*(sqrt(3)/3)", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 2*x);PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};M: Circle;IsDiameter(Abs(LineSegmentOf(A, B)), M);IsTangent(Directrix(G), M);Area(M) = 16*pi", "coords": {} }, { "index": 1320, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "up" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/test/parabola/problem_1320.png", "answer": "x^2=4*y", "fact_expr": "C: Parabola;O: Origin;Vertex(C) = O;PointOnCurve(Focus(C), yAxis);G: Point;Coordinate(G) = (2, 1);PointOnCurve(G, C)", "coords": { "G": [ 2.0, 1.0 ] } }, { "index": 1321, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1322, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1322.png", "answer": "(pm*sqrt(3),0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 4)", "coords": {} }, { "index": 1323, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1323.png", "answer": "(0,sqrt(3)-1]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;LeftVertex(G) = A;B: Point;UpperVertex(G) = B;F: Point;RightFocus(G) = F;MidPoint(LineSegmentOf(A,B)) = M;M: Point;DotProduct(VectorOf(M, F), 2*VectorOf(M, A)) + VectorOf(B, F)^2 >= 0", "coords": {} }, { "index": 1324, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1324.png", "answer": "(4, 2)", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y = 2);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1325, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/test/ellipse/problem_1325.png", "answer": "{16/3,3}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 1326, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1326.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;Coordinate(Focus(G)) = (1,0)", "coords": {} }, { "index": 1327, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1327.png", "answer": "(0,1/16)", "fact_expr": "G: Parabola;F: Point;Expression(G) = (y = 4*x^2);Focus(G) = F", "coords": {} }, { "index": 1328, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1328.png", "answer": "8", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;p: Real;Expression(G) = (x^2/(12 - m) - y^2/(m + 4) = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 1329, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1329.png", "answer": "8*x - y - 15 = 0", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 16*x);Coordinate(P) = (2, 1);PointOnCurve(P, H);Intersection(H, G) = {A, B};VectorOf(P, A) + VectorOf(P, B) = 0", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 1330, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1330.png", "answer": "1/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Circle;e: Number;a > b;b > 0;Expression(H) = (y^2 + (x - 2)^2 = 1);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H);Eccentricity(G) = e", "coords": {} }, { "index": 1331, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1331.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P,G);Distance(P, Focus(G)) = Distance(P, yAxis)*2", "coords": {} }, { "index": 1332, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1333, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_1333.png", "answer": "-21/5", "fact_expr": "A: Point;B: Point;G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Coordinate(B) = (0, b);Eccentricity(G)=sqrt(5)/2;Focus(G)={F1,F2};LeftVertex(G)=A;PointOnCurve(P, LineSegmentOf(A, B))", "coords": {} }, { "index": 1334, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1334.png", "answer": "8", "fact_expr": "G: Hyperbola;k: Number;H: Parabola;Expression(G) = (x^2 - y^2 = k);Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 1335, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1336, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_1336.png", "answer": "x^2-y^2/4=1", "fact_expr": "C: Hyperbola;Expression(Asymptote(C)) = (y = pm*2*x)", "coords": {} }, { "index": 1337, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1337.png", "answer": "4", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;Distance(Focus(G),Asymptote(G))=sqrt(3)", "coords": {} }, { "index": 1338, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1339, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1340.png", "answer": "x^2/5-y^2/20=1", "fact_expr": "l: Line;Expression(l) = (y = 2*x + 10);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(OneOf(Focus(G)), l);IsParallel(l, OneOf(Asymptote(G)))", "coords": {} }, { "index": 1341, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1342, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1342.png", "answer": "(1, sqrt(2)]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 2 = 0);IsIntersect(Asymptote(G),H)", "coords": {} }, { "index": 1343, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1343.png", "answer": "2", "fact_expr": "C: Ellipse;a: Real;O: Origin;F: Point;B: Point;a > sqrt(3);Expression(C) = (y^2/3 + x^2/a^2 = 1);RightFocus(C) = F;UpperVertex(C) = B;Perimeter(TriangleOf(O, F, B)) = sqrt(3) + 3", "coords": {} }, { "index": 1344, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_1344.png", "answer": "y=pm*(1/2)*x", "fact_expr": "C: Hyperbola;a: Number;b: Number;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(5)/2", "coords": {} }, { "index": 1345, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1345.png", "answer": "-1/4", "fact_expr": "C: Ellipse;O: Origin;D: Point;B: Point;A: Point;Expression(C) = (x^2/4 + y^2 = 1);PointOnCurve(A,C);PointOnCurve(B,C);MidPoint(LineSegmentOf(A, B)) = D;Slope(LineOf(O,D))=1", "coords": {} }, { "index": 1346, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1346.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x0, sqrt(2));x0:Number;PointOnCurve(A,G);Distance(A, Focus(G)) = Distance(A, yAxis) * 3", "coords": {} }, { "index": 1347, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1347.png", "answer": "16/3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1348, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1348.png", "answer": "[3/2, +oo)", "fact_expr": "Q: Point;m: Real;Coordinate(Q) = (m, 0);P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);RightVertex(G) = P;WhenMin(Length(LineSegmentOf(P, Q)))", "coords": {} }, { "index": 1349, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -1.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1349.png", "answer": "x^2 + y^2 - 4*x + 2*y + 1 = 0", "fact_expr": "P: Point;A: Point;G: Circle;M: Point;Expression(G) = (2*y - 4*x + x^2 + y^2 - 11 = 0);PointOnCurve(P, G);Center(G) = A;MidPoint(LineSegmentOf(P, A)) = M", "coords": {} }, { "index": 1350, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1350.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 1351, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_1351.png", "answer": "x^2/4-y^2/15=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 1352, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1352.png", "answer": "1/4", "fact_expr": "A: Point;M: Point;G: Hyperbola;a: Real;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/a = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;LeftVertex(G) = A;IsPerpendicular(OneOf(Asymptote(G)), LineOf(A, M));m:Number", "coords": {} }, { "index": 1353, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1354, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1355, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1356, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1356.png", "answer": "x^2/7-y^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2 + x^2/m = 1);m: Number;H: Ellipse;Expression(H) = (x^2/12 + y^2/4 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 1357, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1357.png", "answer": "", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/36 - y^2/m = 1);FocalLength(G) = 18", "coords": {} }, { "index": 1358, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1359, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1359.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;A1: Point;B: Point;A2: Point;C: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;LeftVertex(G) = A1;RightVertex(G) = A2;L:Line;PointOnCurve(F, L);IsPerpendicular(L,xAxis);Intersection(L, G) = {B, C};IsPerpendicular(LineSegmentOf(A1, B), LineSegmentOf(A2, C))", "coords": {} }, { "index": 1360, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1360.png", "answer": "1/9", "fact_expr": "A: Point;M: Point;G: Hyperbola;a: Real;H: Parabola;p: Number;m: Number;Expression(G) = (-y^2 + x^2/a = 1);p>0;m > 0;Expression(H) = (y^2 = 2*(p*x));Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;LeftVertex(G) = A;IsParallel(OneOf(Asymptote(G)), LineOf(A, M))", "coords": {} }, { "index": 1361, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1362, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1363, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1363.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);c: Number;FocalLength(G) = 2*c;Distance(Focus(G),Asymptote(G))=c/2", "coords": {} }, { "index": 1364, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1365, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1365.png", "answer": "2\nx=-1", "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(Focus(G)) = (1, 0)", "coords": {} }, { "index": 1366, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 8.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1366.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/8 + y^2/3 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1367, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1367.png", "answer": "9*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;G: Ellipse;Expression(G) = (4*x^2/9 + y^2/b^2 = 1);b: Number;RightVertex(G) = F;l: Line;Directrix(C) = l;A: Point;PointOnCurve(A, C) = True;PointOnCurve(A, LineSegmentOf(A, B)) = True;IsPerpendicular(LineSegmentOf(A, B), l) = True;FootPoint(LineSegmentOf(A, B), l) = B;B: Point;Slope(LineOf(B, F)) = k_BF;k_BF: Number;k_BF = -sqrt(3)", "coords": {} }, { "index": 1368, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1368.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(F1, P)) = M;O: Origin;Abs(LineSegmentOf(O, M)) = 3", "coords": {} }, { "index": 1369, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1369.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: LineSegment;L:LineSegment;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(F,H);PointOnCurve(F,L);OneOf(Focus(G)) = F ;F: Point;L1:Line;L2:Line;Asymptote(G)={L1,L2};IsPerpendicular(H,L1);IsPerpendicular(L,L2);H + L = a", "coords": {} }, { "index": 1370, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1370.png", "answer": "y^2=16*x", "fact_expr": "Vertex(H) = O;O: Origin;G: Hyperbola;Expression(G) = (x^2/12 - y^2/4 = 1);RightFocus(G) = Focus(H);H: Parabola", "coords": {} }, { "index": 1371, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7841060161590576, "b": 0.3689577579498291, "orientation": "horizontal" }, "optimization": { "final_loss": 5.885034397579147e-07, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_1371.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;a: Number;b: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-sqrt(3)/2, 0);Coordinate(F2) = (sqrt(3)/2, 0);Focus(G) = {F1, F2};Quadrant(P) = 1;PointOnCurve(P, G);Tan(AngleOf(P, F1, F2)) = 1/2;Tan(AngleOf(P, F2, F1)) = -2", "coords": { "F1": [ -0.8660254037844386, 0.0 ], "F2": [ 0.8660254037844386, 0.0 ] } }, { "index": 1372, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1372.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;H: Ellipse;Expression(G) = (x^2/8 - y^2/b^2 = 1);Expression(H) = (x^2/2 + y^2 = 1);Directrix(G)=Directrix(H)", "coords": {} }, { "index": 1373, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1374, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1374.png", "answer": "{(-sqrt(2)/2,sqrt(30)/6),(-sqrt(2)/2,-sqrt(30)/6)}", "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 3*VectorOf(F2, B)", "coords": {} }, { "index": 1375, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1376, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1376.png", "answer": "4*sqrt(5)/5", "fact_expr": "G: Parabola;l1:Line;l2:Line;P:Point;Expression(G) = (y^2 = 4*x);Expression(l1)=(2*x-y+2=0);Expression(l2)=(x=-1);PointOnCurve(P,G)", "coords": {} }, { "index": 1377, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1377.png", "answer": "4/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/m = 1);m: Number;Radius(H) = Eccentricity(G);Center(H) = RightFocus(G);H: Circle;IsTangent(H, OneOf(Asymptote(G))) = True", "coords": {} }, { "index": 1378, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_1378.png", "answer": "9", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 - y^2/m = 1);Eccentricity(G) = 5/4", "coords": {} }, { "index": 1379, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1379.png", "answer": "2*sqrt(42)/7", "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;Q: Point;l1: Line;l2: Line;Expression(G) = (x^2/16 + y^2/8 = 1);Expression(H) = (y^2 + (x - 1)^2 = 1);PointOnCurve(Q, G);TangentOfPoint(Q, H) = {l1, l2};TangentPoint(l1,H)=A;TangentPoint(l2,H)=B", "coords": {} }, { "index": 1380, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1380.png", "answer": "2", "fact_expr": "H: Line;Expression(H) = (y = b + x);b: Number;G: Parabola;Expression(G) = (x^2 = 2*y);Intersection(H, G) = {A, B};A: Point;B: Point;O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True", "coords": {} }, { "index": 1381, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1381.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;B: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -8*x);l1:Line;l2:Line;Asymptote(G)={l1,l2};Intersection(l1, Directrix(G)) = A;Intersection(l2, Directrix(G)) = B;Area(TriangleOf(A, B, O)) = 4*sqrt(3)", "coords": {} }, { "index": 1382, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1382.png", "answer": "25/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);H: Circle;Expression(H) = (x^2 + (y + 2)^2 = 1);A: Point;B: Point;IsDiameter(LineSegmentOf(A, B), H)", "coords": {} }, { "index": 1383, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1383.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;p: Number;A: Point;B:Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,Directrix(H))=A;Intersection(L2,Directrix(H))=B;Eccentricity(G)=2;Area(TriangleOf(A,O,B))=sqrt(3)", "coords": {} }, { "index": 1384, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1384.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (x - 2*y = 0)", "coords": {} }, { "index": 1385, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1385.png", "answer": "(1/2,2/3)", "fact_expr": "C: Ellipse;b: Number;a: Number;H: Line;A:Point;B: Point;e: Number;k:Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C)=A;PointOnCurve(A,H);Slope(H)=k;Intersection(H,C)={A,B};RightFocus(C)=F;IsPerpendicular(LineSegmentOf(B,F),xAxis);1/3 b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) =F1;RightFocus(M)=F2;PointOnCurve(P,M);Range(Max(Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))))=[2*c^2,3*c^2];Eccentricity(M) = e;c=sqrt(a^2-b^2)", "coords": {} }, { "index": 1387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1387.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (x^2 = 2*(p*y));Expression(H) = (x^2/3 + y^2/12 = 1);Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 1388, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1388.png", "answer": "sqrt(13)/2", "fact_expr": "F: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F;PointOnCurve(F, l) = True;l: Line;H: Circle;Expression(H) = (x^2 + y^2 = a^2);IsTangent(l, H) = True;TangentPoint(l, H) = A;A: Point;Intersection(l, RightPart(G)) = B;B: Point;VectorOf(F, A) = VectorOf(F, B)/3", "coords": {} }, { "index": 1389, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1389.png", "answer": "(5/4,-3/4)", "fact_expr": "G: Hyperbola;H: Line;P: Point;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x);PointOnCurve(P, RightPart(G));Distance(P, H) = sqrt(2)", "coords": {} }, { "index": 1390, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1390.png", "answer": "4/3", "fact_expr": "l: Line;G: Parabola;E: Point;F: Point;P: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P, E));IsPerpendicular(LineSegmentOf(P, E), l);FootPoint(LineSegmentOf(P, E), l) = E;Inclination(LineOf(E, F)) = ApplyUnit(150, degree)", "coords": {} }, { "index": 1391, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1391.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(G) = F;A: Point;RightVertex(G) = A;B: Point;Coordinate(B) = (0, b);P: Point;MidPoint(LineSegmentOf(A, B)) = P;Abs(LineSegmentOf(P, F)) = sqrt(6)*Abs(LineSegmentOf(O, P));O: Origin", "coords": {} }, { "index": 1392, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1392.png", "answer": "y=pm*(1/2)*x\nsqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 1393, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1393.png", "answer": "2", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M,G);LeftFocus(G)=F1;Distance(M,F1)=6;MidPoint(LineSegmentOf(M,F1))=N", "coords": {} }, { "index": 1394, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_1394.png", "answer": "x^2/9 - y^2/27 = 1", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 24*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 1395, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1396, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1396.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;Expression(G) = (y^2 = 2*x);Coordinate(P)=(x1,y1);Coordinate(Q)=(x2,y2);PointOnCurve(Focus(G), H);Intersection(H, G)={P,Q};x1:Number;x2:Number;y1:Number;y2:Number;x1+x2=3", "coords": {} }, { "index": 1397, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1397.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "l: Line;G: Ellipse;b: Number;F2: Point;B: Point;F1: Point;A1: Point;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, G) = {A1, B};Max(LineSegmentOf(A1, F2) + LineSegmentOf(B, F2)) = 5", "coords": {} }, { "index": 1398, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1398.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Ellipse;P: Point;Q: Point;Expression(G) = (x^2/4 + y^2/1 = 1);Coordinate(P) = (0, 1);PointOnCurve(Q, G)", "coords": { "P": [ 0.0, 1.0 ] } }, { "index": 1399, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1399.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F:Point;OneOf(Focus(G))=F;l:Line;IsParallel(l,Asymptote(G));Intersection(l, yAxis) = P;P:Point;O:Origin;MidPoint(LineSegmentOf(O,P))=Endpoint(ImageinaryAxis(G));PointOnCurve(F,l)", "coords": {} }, { "index": 1400, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -5.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1400.png", "answer": "10", "fact_expr": "C1: Curve;C2: Curve;C3: Curve;P: Point;Q: Point;R: Point;Expression(C1) = (x^2/16 - y^2/9 = 1);Expression(C2) = (y^2 + (x - 5)^2 = 1);Expression(C3) = (y^2 + (x + 5)^2 = 1);PointOnCurve(P, C1);PointOnCurve(Q, C2);PointOnCurve(R, C3)", "coords": {} }, { "index": 1401, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1401.png", "answer": "3*x+4*y-7=0", "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/4 + y^2/3 = 1);P: Point;Coordinate(P) = (1, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, Gamma) = {A, B};IsChordOf(LineSegmentOf(A, B), Gamma);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 1402, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1403, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1403.png", "answer": "2*sqrt(7)/7", "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G), l);Inclination(l) = pi/3;IsDiameter(MajorAxis(G), H);Length(InterceptChord(l, H)) = FocalLength(G)", "coords": {} }, { "index": 1404, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1404.png", "answer": "[2, 3]", "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 = a^2);a: Real;a>0;G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);IsIntersect(H, G)", "coords": {} }, { "index": 1405, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1405.png", "answer": "24", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2:Point;Expression(G) = (x^2/49 + y^2/24 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))", "coords": {} }, { "index": 1406, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1406.png", "answer": "(1, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;Coordinate(H) = (-1, 1);PointOnCurve(H, Directrix(G))", "coords": { "H": [ -1.0, 1.0 ] } }, { "index": 1407, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1407.png", "answer": "(a^2+b^2)/4", "fact_expr": "P: Point;PointOnCurve(P, E);E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;l1: Line;Expression(l1) = (y = (b/a)*x);l2: Line;Expression(l2) = (y = (-b/a)*x);Asymptote(E) = {l1, l2};Z1: Line;Z2: Line;PointOnCurve(P, Z1);PointOnCurve(P, Z2);IsParallel(Z1, l1);IsParallel(Z2, l2);R: Point;Q: Point;Intersection(Z1, l2) = R;Intersection(Z2, l1) = Q", "coords": {} }, { "index": 1408, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1409.png", "answer": "x-y-2=0", "fact_expr": "G: Parabola;M: Point;N: Point;F:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M,G);PointOnCurve(N,G);Negation(M=N);Focus(G)=F;YCoordinate(MidPoint(LineSegmentOf(M,N)))=2;Abs(LineSegmentOf(M,F))+Abs(LineSegmentOf(N,F))=10", "coords": {} }, { "index": 1410, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1410.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);UpperVertex(G) = A;LeftVertex(G) = B", "coords": {} }, { "index": 1411, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1412, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1412.png", "answer": "4", "fact_expr": "A: Point;B: Point;C:Curve;Expression(C)=(y^2=4*x);IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(Focus(C),LineSegmentOf(A,B));Abs(LineSegmentOf(A, B)) = 10", "coords": {} }, { "index": 1413, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1413.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);O: Origin;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F));DotProduct(VectorOf(O, F), VectorOf(O, A)) + DotProduct(VectorOf(P, A), VectorOf(P, B)) = 0", "coords": {} }, { "index": 1414, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1414.png", "answer": "1/3", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;LeftVertex(G) = A;UpperVertex(G) = B;Distance(F, LineOf(A,B)) = (2*b)/sqrt(17)", "coords": {} }, { "index": 1415, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1415.png", "answer": "2*sqrt(6)", "fact_expr": "G:Parabola;P: Point;O: Origin;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(P,G);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 1416, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1416.png", "answer": "2*sqrt(2)/3", "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(E) = A;RightVertex(E) = B;PointOnCurve(P, E);Tan(AngleOf(P, A, B)) = 1/2;Tan(AngleOf(P, B, A)) = 2/9", "coords": {} }, { "index": 1417, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1417.png", "answer": "sqrt(10)/4", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;A: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);AngleOf(F1, A, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(A, F2))", "coords": {} }, { "index": 1418, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1419, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1419.png", "answer": "(-oo,-32]+(0,oo)", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;M: Point;O: Origin;A: Point;B: Point;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (4, 0);PointOnCurve(M,l);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;Negation(A=B);Eccentricity(G) = 2", "coords": { "M": [ 4.0, 0.0 ] } }, { "index": 1420, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1421, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1422, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1422.png", "answer": "4/5\nx=pm*(25/4)", "fact_expr": "G:Curve;Expression(G)=(x^2/25+y^2/9=1)", "coords": {} }, { "index": 1423, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1423.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "F1: Point;F2: Point;Focus(E) = {F1, F2};E: Hyperbola;Expression(E) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, E);O: Origin;Distance(P, O) = b;Sin(AngleOf(P, F2, F1)) = 3*Sin(AngleOf(P, F1, F2))", "coords": {} }, { "index": 1424, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1425, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1425.png", "answer": "sqrt(2) - 1", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, xAxis);OneOf(Intersection(Z, G)) = P;AngleOf(F1, P, F2) = ApplyUnit(45, degree);Eccentricity(G) = e", "coords": {} }, { "index": 1426, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1427, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1428, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1428.png", "answer": "4", "fact_expr": "G: Ellipse;m: Real;m>1;Expression(G) = (y^2 + x^2/m = 1);Length(MajorAxis(G)) = 2*Length(MinorAxis(G))", "coords": {} }, { "index": 1429, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1429.png", "answer": "y^2=4*x", "fact_expr": "H: Ellipse;Expression(H) = (x^2/5 + y^2/4 = 1);F: Point;RightFocus(H) = F;G: Parabola;Focus(G) = F", "coords": {} }, { "index": 1430, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1430.png", "answer": "4", "fact_expr": "G: Parabola;p: Real;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 1431, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1432, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1432.png", "answer": "(x - 2*sqrt(5)) + y^2 = 25", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/4 - y^2/16 = 1);RightFocus(G) = Center(H);Length(InterceptChord(Asymptote(G),H))=6", "coords": {} }, { "index": 1433, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1433.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G) = True;Distance(M, Focus(G)) = 5", "coords": {} }, { "index": 1434, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1434.png", "answer": "10", "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/m = 1);m: Number;Coordinate(OneOf(Focus(G))) = (0,2)", "coords": {} }, { "index": 1435, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.03125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1435.png", "answer": "65/16", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;M:Point;Expression(G) = (x^2 = y/8);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;YCoordinate(M)=2", "coords": {} }, { "index": 1436, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1436.png", "answer": "5/2", "fact_expr": "G: Ellipse;H: Line;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Focus(G) = {F1,F2};PointOnCurve(F1,H);IsPerpendicular(H,xAxis);OneOf(Intersection(H,G)) = P;IsIntersect(G, H)", "coords": {} }, { "index": 1437, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1437.png", "answer": "(x-3/8)^2 + y^2 = 9/64", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;O: Origin;D: Point;Expression(G) = (x^2/4 + y^2 = 1);t: Real;Coordinate(M) = (1, t);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;PerpendicularBisector(LineSegmentOf(A, B)) = l;IsPerpendicular(LineSegmentOf(O, D), l);FootPoint(LineSegmentOf(O, D), l) = D", "coords": {} }, { "index": 1438, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1438.png", "answer": "3*sqrt(2)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;PointOnCurve(N, Directrix(C));N: Point;L: Line;PointOnCurve(N, L) ;IsPerpendicular(LineSegmentOf(N, F), L);Intersection(L, yAxis) = M;M: Point;PointOnCurve(E, C);E: Point;2*VectorOf(N, E) = VectorOf(N, F) + VectorOf(N, M)", "coords": {} }, { "index": 1439, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1439.png", "answer": "sqrt(3)/2", "fact_expr": "E: Ellipse;b: Number;a: Number;M: Point;A: Point;B: Point;C: Point;D: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MajorAxis(E)) = {A, B};Endpoint(MinorAxis(E)) = {C, D};Abs(LineSegmentOf(M, A))/Abs(LineSegmentOf(M, B)) = 2;Max(TriangleOf(M, A, B)) = 8;Min(TriangleOf(M, C, D)) = 1", "coords": {} }, { "index": 1440, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1440.png", "answer": "7*sqrt(2)/8", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x - y - 2 = 0)", "coords": {} }, { "index": 1441, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1441.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3", "coords": {} }, { "index": 1442, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1442.png", "answer": "sqrt(2)\n(3/4)*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;A: Point;Coordinate(A) = (0, 2);B: Point;MidPoint(LineSegmentOf(F, A)) = B;PointOnCurve(B, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 1443, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1443.png", "answer": "(0,(sqrt(6)-sqrt(2))/4)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,G);Intersection(G, C) = {A, B};AngleOf(A, F, B) >= ApplyUnit(150, degree);O:Origin", "coords": {} }, { "index": 1444, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1444.png", "answer": "sqrt(13) - 2", "fact_expr": "G: Hyperbola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/5 = 1);Coordinate(A) = (0, 2);PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(A, B)) = 2;RightFocus(G) = F", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 1445, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1445.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Line;M: Point;F2: Point;N: Point;Expression(G) = (x^2/4 - y^2/3 = 1);RightFocus(G) = F2;F1: Point;LeftFocus(G) = F1;PointOnCurve(F1, H);Intersection(H, LeftPart(G)) = {M, N}", "coords": {} }, { "index": 1446, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1446.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;F: Point;E: Point;O: Origin;P: Point;l1: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2/4);Coordinate(F) = (-c, 0);LeftFocus(G) = F;TangentOfPoint(F, H) = l1;TangentPoint(l1,H)=E;Intersection(OverlappingLine(LineSegmentOf(F,E)),RightPart(G))=P;VectorOf(O,E)=(VectorOf(O,F)+VectorOf(O,P))/2;c:Number;c>0", "coords": {} }, { "index": 1447, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1447.png", "answer": "5/3", "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 2;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1448, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1448.png", "answer": "-2", "fact_expr": "C: Parabola;G: Circle;M: Point;A: Point;B: Point;x0:Number;y0:Number;Expression(C) = (x^2 = 8*y);Coordinate(M) = (x0, y0);PointOnCurve(M,LineOf(M,A));PointOnCurve(M,LineOf(M,B));TangentPoint(LineOf(M,A),C)=A;TangentPoint(LineOf(M,B),C)=B;IsDiameter(LineSegmentOf(A,B),G);PointOnCurve(M,G)", "coords": {} }, { "index": 1449, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1449.png", "answer": "-4", "fact_expr": "G: Parabola;P: Point;Q: Point;A: Point;l1:Line;l2:Line;Expression(G) = (x^2 = 2*y);PointOnCurve(P,G);PointOnCurve(Q,G);XCoordinate(P)=4;XCoordinate(Q)=-2;TangentOnPoint(P,G)=l1;TangentOnPoint(Q,G)=l2;Intersection(l1,l2)=A", "coords": {} }, { "index": 1450, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1450.png", "answer": "8", "fact_expr": "H: Parabola;Expression(H) = (x^2 = p*y);p: Number;G: Hyperbola;Expression(G) = (-x^2 + y^2/3 = 1);Focus(H) = UpperFocus(G)", "coords": {} }, { "index": 1451, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1451.png", "answer": "(-oo, -sqrt(3)/2) + (sqrt(3)/2, +oo)", "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = -1);O: Origin;PointOnCurve(O, l);NumIntersection(l, G) = 2", "coords": {} }, { "index": 1452, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1452.png", "answer": "(0,pm*3)", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1:Point;F2:Point;Distance(P,F1)*Distance(P,F2) = m;WhenMax(m);m:Number;Focus(G)={F1,F2}", "coords": {} }, { "index": 1453, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1453.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1)", "coords": {} }, { "index": 1454, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1454.png", "answer": "x^2-y^2/3=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;l1: Line;l2: Line;F: Point;O: Origin;P: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(Q, l1);Expression(l1)=(y = -x*b/a);PointOnCurve(P, l2);Expression(l2)=(y = x*(b/a));2*VectorOf(F, Q) = VectorOf(F, P);DotProduct(VectorOf(F, P), VectorOf(O, Q)) = 0;Area(TriangleOf(F, O, P)) = sqrt(3)", "coords": {} }, { "index": 1455, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1456, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1456.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);M: Point;Coordinate(M) = (2, 2);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 1457, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1457.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);Distance(M, Focus(G)) = 5;C: Hyperbola;Expression(C) = (x^2 - y^2/b^2 = 1);b: Number;b>0;A: Point;LeftVertex(C) = A;IsPerpendicular(OneOf(Asymptote(C)), LineOf(A, M))", "coords": {} }, { "index": 1458, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1458.png", "answer": "[sqrt(3)/2,1)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;P: Point;F2: Point;LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);AngleOf(F1,P,F2)=ApplyUnit(120,degree)", "coords": {} }, { "index": 1459, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1460, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.9902384281158447, "b": 1.827926735623088, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.2790036214234405e-06, "converged": true, "iterations": 106 }, "output_path": "results/test/ellipse/problem_1460.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;RightVertex(C) = A;B: Point;UpperVertex(C) = B;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, C);WhenMin(Length(LineSegmentOf(A, B)))", "coords": { "P": [ 2.0, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 1461, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1462, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1462.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (2, 3)", "coords": { "A": [ 2.0, 3.0 ] } }, { "index": 1463, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1463.png", "answer": "sqrt(3)-1", "fact_expr": "C: Ellipse;b: Number;a: Number;F: Point;P: Point;O: Origin;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(P, C);Quadrant(P)=1;AngleOf(F, P, O) = pi/6;Abs(LineSegmentOf(P, F)) = sqrt(3)*Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 1464, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1464.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a^2 = 1);a: Number;a>3;F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) ;PointOnCurve(F1, LineSegmentOf(A, B))", "coords": {} }, { "index": 1465, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1466, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1466.png", "answer": "sqrt(3)/3", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;D: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(A, C);PointOnCurve(B, C);Focus(C) = F;AngleOf(A, F, B) = ApplyUnit(120, degree);MidPoint(LineSegmentOf(A, B))=D;Distance(D, Directrix(C)) = d;d:Number", "coords": {} }, { "index": 1467, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1467.png", "answer": "x = 2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)", "coords": {} }, { "index": 1468, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1468.png", "answer": "4*sqrt(7)", "fact_expr": "G: Ellipse;H: Line;F1: Point;A:Point;C:Circle;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (3*x + 4*y - 21 = 0);LeftFocus(G)=F1;Center(C)=F1;RightVertex(G)=A;PointOnCurve(A,C)", "coords": {} }, { "index": 1469, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1469.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "e: Number;Eccentricity(H) = e;e = 1/2;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = Focus(G);H: Ellipse", "coords": {} }, { "index": 1470, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1470.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1)", "coords": {} }, { "index": 1471, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1471.png", "answer": "16", "fact_expr": "l: Line;G: Parabola;M: Point;N: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (2, 0);Coordinate(N) = (3, 1);PointOnCurve(M, l);PointOnCurve(N, l);Intersection(l,G)={A,B}", "coords": { "M": [ 2.0, 0.0 ], "N": [ 3.0, 1.0 ] } }, { "index": 1472, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -3.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1472.png", "answer": "y^2=12*x", "fact_expr": "l: Line;F: Circle;G:Circle;Expression(F) = (y^2 + (x - 3)^2 = 4);Expression(l)=(x=-1);IsOutTangent(F,G);IsTangent(l,G)", "coords": {} }, { "index": 1473, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1474, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1474.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;D: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(G) = A;UpperVertex(G) = B;PointOnCurve(P, G);Intersection(LineSegmentOf(A, P), yAxis) = C;Intersection(LineSegmentOf(B, P), xAxis) = D;Abs(LineSegmentOf(A, D))*Abs(LineSegmentOf(B, C)) = a^2", "coords": {} }, { "index": 1475, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1475.png", "answer": "x^2/4-y^2/3=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (2, sqrt(3));Coordinate(F) = (-sqrt(7), 0);PointOnCurve(H, OneOf(Asymptote(G)));OneOf(Focus(G)) = F", "coords": { "H": [ 2.0, 1.7320508075688772 ], "F": [ -2.6457513110645907, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1476, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 9.0, "b": 5.196152422706632, "major_axis": "x", "x_coef": 81.0, "y_coef": 27.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1476.png", "answer": "36", "fact_expr": "G: Ellipse;Expression(G) = (x^2/81 + y^2/27 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);P: Point;Q: Point;Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 1477, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1477.png", "answer": "4*sqrt(3)/3", "fact_expr": "l: Line;G: Parabola;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Inclination(l) = pi/3;Intersection(l, G) = {A, B}", "coords": {} }, { "index": 1478, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1478.png", "answer": "sqrt(15)", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (1, 1);IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 1479, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1479.png", "answer": "1/(Cos(alpha)-Sin(alpha))", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;IsDiameter(LineSegmentOf(F1,F2),H);H: Circle;Intersection(H,RightPart(G)) = P;P: Point;AngleOf(P, F1, F2) = alpha;alpha: Number", "coords": {} }, { "index": 1480, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1480.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 1481, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1481.png", "answer": "4", "fact_expr": "G: Ellipse;H: Line;M: Point;N: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {M, N};Area(InscribedCircle(TriangleOf(M, N, F2))) = pi", "coords": {} }, { "index": 1482, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1482.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;P: Point;F: Point;O: Origin;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 1483, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1483.png", "answer": "-4/3", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;O: Origin;E: Point;P: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (b + x - 2*y = 0);Eccentricity(G)= sqrt(3)/3;Intersection(H, G) = {P, Q};MidPoint(LineSegmentOf(P,Q))=E", "coords": {} }, { "index": 1484, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1485, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1485.png", "answer": "ApplyUnit(90,degree)", "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (y = x - 2);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1486, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1486.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);P: Point;PointOnCurve(P, G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};l1: Line;l2: Line;PointOnCurve(P, l1);PointOnCurve(P, l2);IsParallel(L1, l1);IsParallel(L2, l2);M: Point;N: Point;Intersection(l1, xAxis) = M;Intersection(l2, xAxis) = N;O: Origin", "coords": {} }, { "index": 1487, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1487.png", "answer": "sqrt(2)*27/16", "fact_expr": "G: Parabola;P: Point;A: Point;B: Point;C: Point;l1: Line;l2: Line;Expression(G) = (y = x^2);Coordinate(P) = (0, 1);TangentOfPoint(C, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;VectorOf(C, P) = VectorOf(C, A)/3 + (2/3)*VectorOf(C, B)", "coords": { "P": [ 0.0, 1.0 ] } }, { "index": 1488, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1488.png", "answer": "x^2/18+y^2/8=1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (0, -1);Eccentricity(C) = sqrt(3)/3;Center(G)=M;IsIntersect(G,C);Max(Radius(G))=sqrt(15)", "coords": { "M": [ 0.0, -1.0 ] } }, { "index": 1489, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1489.png", "answer": "x=-1/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)", "coords": {} }, { "index": 1490, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1490.png", "answer": "12", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 1491, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1491.png", "answer": "[-5,5]", "fact_expr": "G: Ellipse;P: Point;x1:Number;y1:Number;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": {} }, { "index": 1492, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1493, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 1494, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1494.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Intersection(l, G) = {P, Q};XCoordinate(MidPoint(LineSegmentOf(P, Q))) = 3", "coords": {} }, { "index": 1495, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1496, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1496.png", "answer": "y = pm*x", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/8 + x^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (4, 0)", "coords": {} }, { "index": 1497, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1497.png", "answer": "(-1/2,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -2*x)", "coords": {} }, { "index": 1498, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1498.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F2: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {A, B};Abs(LineSegmentOf(A, F2)) = 6;Abs(LineSegmentOf(B, F2)) = 10;Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 1499, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1500, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1500.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;c:Number;H: Circle;F1: Point;M: Point;F2: Point;a>0;b>0;c>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;Center(H)=F2;Radius(H)=c;OneOf(Intersection(H,RightPart(G)))=M;IsTangent(LineOf(F1,M),H)", "coords": {} }, { "index": 1501, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1501.png", "answer": "a^2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;R: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);PointOnCurve(P, H);IsParallel(H, RealAxis(G));L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(H, L1) = R;Intersection(H, L2) = Q", "coords": {} }, { "index": 1502, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1502.png", "answer": "sqrt(3) - 1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;F2: Point;M: Point;F1: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;Center(H) = F2;PointOnCurve(Center(G), H);Intersection(H, G) = {M, N};LeftFocus(G) = F1;PointOnCurve(F1, LineOf(M, F1));IsTangent(LineOf(M, F1), H)", "coords": {} }, { "index": 1503, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1503.png", "answer": "4*sqrt(6)", "fact_expr": "l: Line;G: Parabola;M: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (2, 0);PointOnCurve(M, l);Slope(l) = 1;Intersection(l, G) = {A, B}", "coords": { "M": [ 2.0, 0.0 ] } }, { "index": 1504, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1504.png", "answer": "(x - 2)^2 + (y + 1)^2 = 1", "fact_expr": "P: Point;Coordinate(P) = (4, -2);G: Circle;Expression(G) = (x^2 + y^2 = 4);Z: Point;PointOnCurve(Z, G)", "coords": {} }, { "index": 1505, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1505.png", "answer": "4*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;D: Point;E: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(G,L1)=D;Intersection(G,L2)=E;Area(TriangleOf(O, D, E)) = 4", "coords": {} }, { "index": 1506, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1507, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1507.png", "answer": "66/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/36 = 1);P: Point;PointOnCurve(P, G);Distance(P, RightDirectrix(G)) = 17/2", "coords": {} }, { "index": 1508, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1508.png", "answer": "5/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2 + y^2/5 = 1);A: Point;RightVertex(C) = A;P: Point;PointOnCurve(P, C) = True", "coords": {} }, { "index": 1509, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/test/hyperbola/problem_1509.png", "answer": "x^2/2-y^2/4=1", "fact_expr": "G: Hyperbola;P: Point;O: Origin;Center(G)=O;Coordinate(P) = (1, -2);PointOnCurve(Focus(G), xAxis);PointOnCurve(P, G);Expression(OneOf(Asymptote(G)))=(y=sqrt(2)*x)", "coords": { "P": [ 1.0, -2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1510, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1510.png", "answer": "(1, 1)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2);Expression(H) = (2*x - y = 4);Z: Point;PointOnCurve(Z, G);WhenMin(Distance(Z, H))", "coords": {} }, { "index": 1511, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1512, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/test/hyperbola/problem_1512.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);OneOf(Focus(G)) = F;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1513, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1514, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1514.png", "answer": "{0,1/8}", "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);G: Parabola;Expression(G) = (y^2 = x);NumIntersection(H, G) = 1;k: Real", "coords": {} }, { "index": 1515, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1515.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/12 + x^2/a^2 = 1);a: Number;a>0;Inclination(OneOf(Asymptote(G))) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1516, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1516.png", "answer": "2*sqrt(3)\ny=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1)", "coords": {} }, { "index": 1517, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1518, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1518.png", "answer": "5/3", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;P: Point;PointOnCurve(P, LeftPart(E));Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));G: Circle;Expression(G) = (x^2 + y^2 = a^2);IsTangent(LineOf(P, F2), G)", "coords": {} }, { "index": 1519, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1519.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (1, 4);l: Line;PointOnCurve(P,l) = True;Intersection(l,G) = {A,B};A: Point;B: Point;MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 1.0, 4.0 ] } }, { "index": 1520, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1520.png", "answer": "3/2", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;P: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (-1, 1/2);PointOnCurve(P, l);Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ -1.0, 0.5 ] } }, { "index": 1521, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1521.png", "answer": "(x-2)^2 + (y - 4*sqrt(2))^2 = 36", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);H: Circle;PointOnCurve(Center(H), G);PointOnCurve(Vertex(G), H);PointOnCurve(Focus(G), H);Quadrant(Center(H)) = 1", "coords": {} }, { "index": 1522, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1522.png", "answer": "1", "fact_expr": "l: Line;Expression(l) = (x - y + 1 = 0);C: Parabola;Expression(C) = (x^2 = 2*y);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;PointOnCurve(P, l);M: Point;N: Point;PointOnCurve(M, C);PointOnCurve(N, C);IsParallel(VectorOf(M, N), VectorOf(A, B));Abs(VectorOf(M, N)) < Abs(VectorOf(A, B))", "coords": {} }, { "index": 1523, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1523.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Length(ImageinaryAxis(G)) = 2;FocalLength(G) = 2*sqrt(3)", "coords": {} }, { "index": 1524, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1524.png", "answer": "2*sqrt(3)-2", "fact_expr": "G: Hyperbola;H: Circle;P: Point;F: Point;N: Point;O: Origin;T: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Expression(H) = (x^2 + y^2 = 4);LeftFocus(G)=F;TangentOfPoint(F,H)=LineOf(F,P);Intersection(LineOf(F,P),RightPart(G))=P;TangentPoint(LineOf(F,P),H)=T;MidPoint(LineSegmentOf(F,P)) = N", "coords": {} }, { "index": 1525, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1525.png", "answer": "-1/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;RightFocus(C) = F;G: Line;Expression(G) = (y = k*x + 1);k: Real;A: Point;B: Point;Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 1526, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1526.png", "answer": "9", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G)=F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 1527, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1528.png", "answer": "7", "fact_expr": "l1: Line;b: Number;b>3;Expression(l1) = (x - sqrt(3)*y + b = 0);C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l2: Line;Directrix(C) = l2;A: Point;PointOnCurve(A, C);d1: Number;d2: Number;Distance(A, l1) = d1;Distance(A, l2) = d2;WhenMin(d1+d2);d1 = 2*d2", "coords": {} }, { "index": 1529, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1529.png", "answer": "2*x+4*y-3=0", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (1/2, 1/2);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 1530, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1531, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1531.png", "answer": "(3/2, sqrt(15)/2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;PointOnCurve(M, C);Quadrant(M) = 1;Abs(LineSegmentOf(M, F1)) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 1532, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1533, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1534, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1534.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Vertex(TriangleOf(A,B,C)),G);IsPerpendicular(LineSegmentOf(A,B),xAxis);IsParallel(LineSegmentOf(A,C),xAxis)", "coords": {} }, { "index": 1535, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1535.png", "answer": "1/7", "fact_expr": "P: Point;A: Point;G: Ellipse;b: Number;a: Number;B: Point;a > b;b > 0;e: Number;alpha: Number;beta: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e ;e = 1/2;LeftVertex(G) = A;RightVertex(G) = B;PointOnCurve(P, G);Negation(P = A);Negation(P = B);Inclination(LineOf(P, A)) = alpha;Inclination(LineOf(P, B)) = beta", "coords": {} }, { "index": 1536, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1536.png", "answer": "25/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;N: Point;F2: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, LeftPart(C));L:Line;PointOnCurve(M, L);IsPerpendicular(L,OneOf(Asymptote(C)));FootPoint(L,OneOf(Asymptote(C)))=N;Min(Abs(LineSegmentOf(M,F2))+Abs(LineSegmentOf(M,N)))=10", "coords": {} }, { "index": 1537, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1537.png", "answer": "2*sqrt(2)/3", "fact_expr": "l: Line;C: Parabola;B: Point;P: Point;A: Point;Expression(C) = (y^2 = 8*x);Coordinate(P) = (-2, 0);PointOnCurve(P, l);Intersection(l, C) = {A, B};Quadrant(A) = 1;Quadrant(B) = 1;MidPoint(LineSegmentOf(P,B)) = A", "coords": { "P": [ -2.0, 0.0 ] } }, { "index": 1538, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1538.png", "answer": "{3,1/3}", "fact_expr": "G: Hyperbola;m: Number;n: Number;Expression(G) = (-y^2/n + x^2/m = 1);Eccentricity(G) = 2;Negation(m*n=0)", "coords": {} }, { "index": 1539, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1540, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1540.png", "answer": "-b^2/a^2", "fact_expr": "P: Point;A: Point;G: Ellipse;a: Number;b: Number;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MinorAxis(G)) = {A, B};PointOnCurve(P, G);Negation(P = A);Negation(P = B)", "coords": {} }, { "index": 1541, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1542, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1542.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;H: Line;Expression(H) = (m - m*y + x = 0);PointOnCurve(Focus(G), H);m: Number;Intersection(H, G) = {A, B};O: Origin;A: Point;B: Point;Area(TriangleOf(O, A, B)) = 2*sqrt(2)", "coords": {} }, { "index": 1543, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1543.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F1: Point;LeftFocus(G) = F1;PointOnCurve(F1, LineOf(A, B)) = True;Inclination(LineOf(A, B)) = pi/6;A: Point;B: Point;Intersection(LineOf(A, B), LeftPart(G)) = A;Intersection(LineOf(A, B), RightPart(G)) = B;IsChordOf(LineSegmentOf(A, B), G) = True", "coords": {} }, { "index": 1544, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1545, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1545.png", "answer": "6", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = -8*x);PointOnCurve(P, G);Distance(P, yAxis) = 4", "coords": {} }, { "index": 1546, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1547, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1548, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1548.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;y1: Number;y2: Number;Coordinate(A) = (2, y1);Coordinate(B) = (1/2, y2);PointOnCurve(A, G);PointOnCurve(B, G);Quadrant(A) = 1;Quadrant(B) = 4;Abs(LineSegmentOf(A, F)) = 10", "coords": {} }, { "index": 1549, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1549.png", "answer": "y=pm*((4/3)*x)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2, LineOf(P, F1)) = Length(RealAxis(G))", "coords": {} }, { "index": 1550, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1550.png", "answer": "2/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 2);H: Line;PointOnCurve(Focus(G), H) = True;Inclination(H) = ApplyUnit(45, degree);A: Point;B: Point;Intersection(H,G) = {A,B};O: Origin;Center(G) = O", "coords": {} }, { "index": 1551, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1551.png", "answer": "", "fact_expr": "G: Ellipse;F2: Point;B: Point;F1: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)", "coords": {} }, { "index": 1552, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1552.png", "answer": "1", "fact_expr": "l: Line;E: Parabola;F: Point;A: Point;B: Point;Expression(E) = (y^2 = 4*x);Focus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B}", "coords": {} }, { "index": 1553, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1553.png", "answer": "{(2, 3), sqrt(13)}", "fact_expr": "G: Ellipse;C: Curve;m: Real;Expression(G) = (x^2/9 + y^2/4 = 1);m>0;Expression(C) = (Abs(x) + Abs(y) = m);NumIntersection(C, G) = 4", "coords": {} }, { "index": 1554, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1554.png", "answer": "(1,2)", "fact_expr": "G: Hyperbola;a: Number;b: Number;A: Point;B: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);Coordinate(B) = (-1, 2);IsPerpendicular(VectorOf(A,P),VectorOf(B,P));NumIntersection(Asymptote(G),Locus(P))=0", "coords": { "A": [ 1.0, 2.0 ], "B": [ -1.0, 2.0 ] } }, { "index": 1555, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1555.png", "answer": "1", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(F, P1), LineSegmentOf(F, P2))", "coords": {} }, { "index": 1556, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.47213595499958, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1556.png", "answer": "10", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/20 - y^2/5 = 1)", "coords": {} }, { "index": 1557, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1558, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1558.png", "answer": "3*sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);N: Point;Coordinate(N) = (2, 0);H: Line;PointOnCurve(N, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, N))/Abs(LineSegmentOf(B, N)) = 2", "coords": { "N": [ 2.0, 0.0 ] } }, { "index": 1559, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1559.png", "answer": "[sqrt(5)/5,sqrt(3)/3]", "fact_expr": "Gamma: Ellipse;a: Number;b: Number;A: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;lambda:Number;In(lambda,[2,3]);Expression(Gamma) = (x^2/a^2 + y^2/b^2 = 1);LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;PointOnCurve(A, Gamma);PointOnCurve(B, Gamma);DotProduct(VectorOf(A,F1),VectorOf(F1,F2))=0;VectorOf(A,F2)=lambda*VectorOf(F2,B)", "coords": {} }, { "index": 1560, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1560.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 + y^2 = 1);Slope(H)=1;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1561, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1561.png", "answer": "x = -1/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)", "coords": {} }, { "index": 1562, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1562.png", "answer": "y=2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)", "coords": {} }, { "index": 1563, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1563.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x + y + 2 = 0);Intersection(H,xAxis) = A;Intersection(H,yAxis) = B;PointOnCurve(P, G)", "coords": {} }, { "index": 1564, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1565, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1565.png", "answer": "sqrt(11)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/7 + y^2/4 = 1)", "coords": {} }, { "index": 1566, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1566.png", "answer": "pm*(4/3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;O: Origin;H: Circle;A: Point;TangentPoint(LineOf(P, F1), H) = A;Center(H) = O;Radius(H) = a;PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1)))", "coords": {} }, { "index": 1567, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 9.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1567.png", "answer": "2*sqrt(6)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)", "coords": {} }, { "index": 1568, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1569, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1570, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 25.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1570.png", "answer": "(25, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 100*x)", "coords": {} }, { "index": 1571, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1571.png", "answer": "x^2+y^2/2=1", "fact_expr": "G: Ellipse;O: Origin;P: Point;Q:Point;Expression(G) = (x^2/4 + y^2/8 = 1);PointOnCurve(P, G);MidPoint(LineSegmentOf(O,P))=Q", "coords": {} }, { "index": 1572, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1572.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 1573, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1573.png", "answer": "1+sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Inclination(H)=ApplyUnit(45,degree);Intersection(H, RightPart(G)) = M;IsPerpendicular(LineSegmentOf(M,F2),xAxis)", "coords": {} }, { "index": 1574, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1574.png", "answer": "3", "fact_expr": "C: Parabola;F: Point;N: Point;M: Point;l: Line;lambda:Number;lambda>0;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(N, l);VectorOf(F, N) = lambda*VectorOf(F, M);Abs(LineSegmentOf(M, F)) = 4/3", "coords": {} }, { "index": 1575, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1576, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1576.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B}", "coords": {} }, { "index": 1577, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1577.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (-x^2 + y^2/3 = 1);Expression(H) = (x^2 = 8*y)", "coords": {} }, { "index": 1578, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.5, "b": 0.8660254037844386, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1578.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "l: Line;G: Hyperbola;H: Ellipse;P: Point;Expression(G) = (12*x^2 - 4*y^2 = 3);Expression(l) = (y = x + 3);PointOnCurve(P, l);PointOnCurve(P, H);Focus(G) = Focus(H);WhenMin(MajorAxis(H))", "coords": {} }, { "index": 1579, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1579.png", "answer": "sqrt(2)/2", "fact_expr": "A: Point;B: Point;E: Ellipse;Expression(E) = (y^2 + x^2/m = 1);m: Number;m > 1;Vertex(MinorAxis(E)) = {A, B};P: Point;PointOnCurve(P, E) = True;Negation(P = A);Negation(P = B);Slope(LineOf(A, P))*Slope(LineOf(B, P)) = -m/4", "coords": {} }, { "index": 1580, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1580.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(P, G);Distance(P, xAxis) = 2", "coords": {} }, { "index": 1581, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1581.png", "answer": "(-sqrt(10)/2,-sqrt(2)/2)+(-sqrt(2)/2,sqrt(2)/2)+(sqrt(2)/2,sqrt(10)/2)", "fact_expr": "l: Line;C: Hyperbola;Expression(C) = (x^2/2 - y^2 = 1);Expression(l)=(y=k*x+2);NumIntersection(l,C)=2;k:Number", "coords": {} }, { "index": 1582, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1582.png", "answer": "pm*3*y+4*x=0", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G)=(x^2/25+y^2/16=1);Focus(C)=Endpoint(MajorAxis(G));Endpoint(RealAxis(C))=Focus(G)", "coords": {} }, { "index": 1583, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1583.png", "answer": "x=-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 1584, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1584.png", "answer": "(1,sqrt(2)]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 2 = 0);IsIntersect(Asymptote(G),H)", "coords": {} }, { "index": 1585, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1585.png", "answer": "45/2", "fact_expr": "A: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);LeftVertex(G) = A;B: Point;PointOnCurve(B, G) = True;PointOnCurve(C, yAxis) = True;C: Point;Coordinate(C) = (0, 4)", "coords": { "C": [ 0.0, 4.0 ] } }, { "index": 1586, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1586.png", "answer": "4/5", "fact_expr": "G: Ellipse;m: Number;m>0;Expression(G)=(y^2/9 + x^2/m^2 = 1);Coordinate(OneOf(Focus(G)))=(4,0)", "coords": {} }, { "index": 1587, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1587.png", "answer": "12*sqrt(6)", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 6*sqrt(6));RightFocus(C) = F;PointOnCurve(P, LeftPart(C));WhenMin(Perimeter(TriangleOf(A,P,F)))", "coords": { "A": [ 0.0, 14.696938456699067 ] } }, { "index": 1588, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1588.png", "answer": "(-6, -6)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*p*y);p: Number;p>0;F: Point;Coordinate(F) = (0, -3);Focus(G) = F;H: Line;Expression(H) = (y = x)", "coords": { "F": [ 0.0, -3.0 ] } }, { "index": 1589, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1589.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e ;e: Number;e = 2", "coords": {} }, { "index": 1590, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1590.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;O: Origin;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 1591, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1592, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1593, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -4.0, 0.0 ], "radius": 13.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1593.png", "answer": "x^2/64+y^2/48=1", "fact_expr": "M: Circle;Expression(M) = (y^2 + (x - 4)^2 = 169);N: Circle;Expression(N) = (y^2 + (x + 4)^2 = 9);P: Circle;IsInTangent(P,M) = True;IsOutTangent(P,N) = True;P1: Point;Center(P) = P1", "coords": {} }, { "index": 1594, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 6.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1594.png", "answer": "-1", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/6 + y^2/3 = 1);Focus(G) = {F1, F2};A: Point;B: Point;LeftVertex(G) = A;UpperVertex(G) = B;P: Point;PointOnCurve(P, LineSegmentOf(A, B)) = True", "coords": {} }, { "index": 1595, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1595.png", "answer": "8", "fact_expr": "C: Ellipse;G: Line;F2: Point;M: Point;N: Point;F1: Point;Expression(C) = (x^2/3 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(F1,G);Intersection(G,C) = {M,N}", "coords": {} }, { "index": 1596, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1596.png", "answer": "5", "fact_expr": "A: Point;Coordinate(A) = (-1, 1);C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);M: Point;PointOnCurve(M, C) = True;F: Point;RightFocus(C) = F", "coords": { "A": [ -1.0, 1.0 ] } }, { "index": 1597, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1598, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1598.png", "answer": "x^2/3-y^2=1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);Focus(C) = Focus(G);Expression(OneOf(Asymptote(C))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 1599, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1600.png", "answer": "(3 + pm*sqrt(5))/4", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 1/2", "coords": {} }, { "index": 1601, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1601.png", "answer": "(1+sqrt(145))/18", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;A: Point;B: Point;PointOnCurve(F2, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(F1, F2));Abs(LineSegmentOf(F1, A)) = 1/2*Abs(LineSegmentOf(F1, B))", "coords": {} }, { "index": 1602, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1602.png", "answer": "(0,1/2)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2)", "coords": {} }, { "index": 1603, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1604, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1604.png", "answer": "4", "fact_expr": "E: Ellipse;G: Line;m: Number;F1: Point;A: Point;B: Point;Expression(E) = (x^2/25 + y^2/9 = 1);Expression(G) = (x = m);LeftFocus(E) = F1;Intersection(G, E) = {A, B};WhenMax(Perimeter(TriangleOf(F1,A,B)))", "coords": {} }, { "index": 1605, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1606, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1606.png", "answer": "1/3", "fact_expr": "G: Ellipse;b: Number;a: Number;e: Number;H: Line;A: Point;P: Point;B: Point;C: Point;Q: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e ;e = 1/2;PointOnCurve(Center(G), H);Intersection(H, G) = {A, B};Quadrant(A) = 1;Z: Line;PointOnCurve(A, Z);IsPerpendicular(Z, xAxis);Intersection(Z, G) = C;PointOnCurve(A, LineOf(A, P));IsPerpendicular(LineOf(A, P), LineSegmentOf(A, B));Intersection(LineOf(A, P), G) = P;Intersection(LineSegmentOf(B, P), LineSegmentOf(A, C)) = Q", "coords": {} }, { "index": 1607, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "y", "x_coef": 16.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1607.png", "answer": "(0,pm*3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/25 = 1)", "coords": {} }, { "index": 1608, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1609, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1609.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) ={F1,F2};PointOnCurve(P, G);AngleOf(F1,P,F2)=ApplyUnit(60,degree)", "coords": {} }, { "index": 1610, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1610.png", "answer": "16", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(C)=Focus(G);Expression(LeftDirectrix(C))=(x=-1);P: Point;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C))", "coords": {} }, { "index": 1611, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1611.png", "answer": "[2*sqrt(2) - 3, 56/9]", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 1)^2 = 1);G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);L1: Line;L2: Line;TangentPoint(L1, C) = A;TangentPoint(L2, C) = B;A: Point;B: Point;TangentOfPoint(P, C) = {L1, L2}", "coords": {} }, { "index": 1612, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1612.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(G) = 2*sqrt(5);Length(RealAxis(G)) = 2*Length(ImageinaryAxis(G))", "coords": {} }, { "index": 1613, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1613.png", "answer": "y=pm*(x/2)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/4 = 1)", "coords": {} }, { "index": 1614, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1614.png", "answer": "[sqrt(7)/4, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a = 1);a: Number;a>9;H: Line;Expression(H) = (x - sqrt(3)*y = 0);Negation(Distance(OneOf(Vertex(MajorAxis(G))),H) < 2)", "coords": {} }, { "index": 1615, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1615.png", "answer": "5/6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;H: Line;A: Point;B: Point;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 25/12;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 1616, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1616.png", "answer": "sqrt(2)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (m, 0);Coordinate(P) = (m, 2*m);PointOnCurve(P, G);m:Number;m>0", "coords": {} }, { "index": 1617, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1617.png", "answer": "9", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/7 - y^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 8", "coords": {} }, { "index": 1618, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1618.png", "answer": "6", "fact_expr": "G: Ellipse;M: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(M, G);Distance(M, RightDirectrix(G)) = 20/3", "coords": {} }, { "index": 1619, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1620, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1620.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;P: Point;O: Origin;Negation(P=O);PointOnCurve(P, G);H: Line;PointOnCurve(P, H);Z: Parabola;Expression(Z) = (y^2 = 8*p*x);A: Point;B: Point;Intersection(H, Z)={A, B};Q: Point;Intersection(LineOf(O,P), Z)={P, Q}", "coords": {} }, { "index": 1621, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1621.png", "answer": "5", "fact_expr": "l: Line;C: Hyperbola;P: Point;F1: Point;Q: Point;Expression(C) = (x^2/4 - y^2 = 1);PointOnCurve(P, RightPart(C));OneOf(Asymptote(C))=l;Projection(P,l)=Q;LeftFocus(C)=F1", "coords": {} }, { "index": 1622, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/test/hyperbola/problem_1622.png", "answer": "(pm*2, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 1623, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1623.png", "answer": "32/15", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);A: Point;RightVertex(G) = A;RightFocus(G) = F;F: Point;PointOnCurve(F, H) = True;IsParallel(H, OneOf(Asymptote(G))) = True;H: Line;Intersection(H, G) = B;B: Point", "coords": {} }, { "index": 1624, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1624.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "O: Origin;Center(H) = O;H: Ellipse;PointOnCurve(I, H);I: Point;Coordinate(I) = (2, -3);G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G)", "coords": { "I": [ 2.0, -3.0 ] } }, { "index": 1625, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1626, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1627, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5773502691896257, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.3333333333333333 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1627.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 1)", "coords": {} }, { "index": 1628, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1628.png", "answer": "y^2/12-x^2/18=1", "fact_expr": "G: Hyperbola;C:Hyperbola;M: Point;Expression(G) = (x^2/3 - y^2/2 = 1);Coordinate(M) = (2*sqrt(3), 2*sqrt(5));Asymptote(G) = Asymptote(C);PointOnCurve(M,C)", "coords": { "M": [ 3.4641016151377544, 4.47213595499958 ] } }, { "index": 1629, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1629.png", "answer": "3", "fact_expr": "G: Parabola;p: Number;E: Point;O: Origin;M: Point;F: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(M, G);Focus(G) = F;L:Line;PointOnCurve(M, L);IsPerpendicular(L, l);FootPoint(L, l)=E;Abs(LineSegmentOf(E, O)) = Abs(LineSegmentOf(M, F));XCoordinate(M) = 3;Directrix(G)=l", "coords": {} }, { "index": 1630, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1630.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/b^2 - y^2/a^2 = 1);b: Number;a: Number;l1: Line;l2: Line;Asymptote(G) = {l1, l2};IsPerpendicular(l1, l2)", "coords": {} }, { "index": 1631, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1631.png", "answer": "14/3", "fact_expr": "G: Hyperbola;Expression(G) = (5*x^2 - 4*y^2 = 20);P: Point;PointOnCurve(P, G) = True;Distance(P, RightFocus(G)) = 3", "coords": {} }, { "index": 1632, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1632.png", "answer": "[3/5, 3/4]", "fact_expr": "G: Ellipse;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;Coordinate(F) = (0, -3);OneOf(Focus(G)) = F;A: Point;Coordinate(A) = (0, 2);P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)) = 9", "coords": { "F": [ 0.0, -3.0 ], "A": [ 0.0, 2.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1633, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1633.png", "answer": "100/3", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);G: Parabola;Center(H) = Vertex(G);LeftDirectrix(H) = Directrix(G);Intersection(G,RightDirectrix(H)) = {A,B};A: Point;B: Point", "coords": {} }, { "index": 1634, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1634.png", "answer": "sqrt(6)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/5 = 1)", "coords": {} }, { "index": 1635, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1635.png", "answer": "x^2/4-y^2/12=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G)=Focus(H);Eccentricity(G)=2", "coords": {} }, { "index": 1636, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1636.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 2", "coords": {} }, { "index": 1637, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_1637.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;Q: Point;F1: Point;M: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (-sqrt(2), -sqrt(3));Coordinate(Q) = (sqrt(15)/3, sqrt(2));PointOnCurve(P,C);PointOnCurve(Q,C);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M,C);AngleOf(F1, M, F2) = ApplyUnit(120, degree)", "coords": { "P": [ -1.4142135623730951, -1.7320508075688772 ], "Q": [ 1.2909944487358056, 1.4142135623730951 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1638, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1638.png", "answer": "sqrt(2)", "fact_expr": "C1: Hyperbola;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;PointOnCurve(F1, l) = True;Intersection(l, RightPart(C1)) = P;l: Line;P: Point;IsPerpendicular(LineSegmentOf(P, F2), xAxis) = True;Sin(AngleOf(P, F1, F2)) = 1/3", "coords": {} }, { "index": 1639, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1639.png", "answer": "8*sqrt(5)/5", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G) = True;l: Line;Expression(l) = (x + 2*y - 12 = 0)", "coords": {} }, { "index": 1640, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, 2.0 ], "radius": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1640.png", "answer": "y^2/9 + x^2/5 = 1", "fact_expr": "C1: Circle;Expression(C1) = (-4*y + x^2 + y^2 - 32 = 0);B: Point;PointOnCurve(B, C1);A: Point;Coordinate(A) = (0, -2);C: Point;Center(C1) = C;P: Point;Intersection(PerpendicularBisector(LineSegmentOf(A,B)), LineSegmentOf(B, C)) = P;Locus(P) = E;E: Curve", "coords": {} }, { "index": 1641, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1641.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;Z: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;PointOnCurve(F1, Z);IsPerpendicular(Z, xAxis);Intersection(Z, G) = P;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1642, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1642.png", "answer": "1/2\n{((2*sqrt(2))*x-y-4*sqrt(2)=0), ((2*sqrt(2))*x+y-4*sqrt(2)=0)}", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 8*x);F: Point;Focus(E) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, E) = {A, B};VectorOf(A,B) = 3*VectorOf(F,B)", "coords": {} }, { "index": 1643, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1643.png", "answer": "8*x+25*y-58=0", "fact_expr": "M: Point;Coordinate(M) = (1, 2);l: Line;PointOnCurve(M, l);G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 1644, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1645, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 7.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1645.png", "answer": "3", "fact_expr": "G: Ellipse;F1: Point;M: Point;F2: Point;O: Origin;Expression(G) = (x^2/7 + y^2/3 = 1);PointOnCurve(M, G);Focus(G) = {F1, F2};2*LineSegmentOf(M, O) = LineSegmentOf(F1, F2)", "coords": {} }, { "index": 1646, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 4.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1646.png", "answer": "9", "fact_expr": "G: Circle;C:Circle;H: Curve;P: Point;M: Point;N: Point;Expression(H)=(sqrt((x+4)^2+y^2)+sqrt((x-4)^2+y^2)=12);Expression(G) = (y^2 + (x + 4)^2 = 4);Expression(C) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P,H);PointOnCurve(M, G);PointOnCurve(N, C)", "coords": {} }, { "index": 1647, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1647.png", "answer": "1/4", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;p>0;M: Point;Coordinate(M) = (1, m);m: Number;PointOnCurve(M, H);Distance(M, Focus(H)) = 5;G: Hyperbola;Expression(G) = (x^2 - y^2/a = 1);a: Real;A: Point;LeftVertex(G) = A;IsPerpendicular(OneOf(Asymptote(G)), LineOf(A, M))", "coords": {} }, { "index": 1648, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1648.png", "answer": "(4,2)", "fact_expr": "H: Line;Expression(H) = (x - y = 2);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1649, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1650, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1650.png", "answer": "{2,-14}", "fact_expr": "G: Parabola;a: Real;H: Circle;Expression(G) = (y^2 = 2*(a*x));Expression(H) = (y^2 + (x - 3)^2 = 16);IsTangent(Directrix(G),H);Negation(a=0)", "coords": {} }, { "index": 1651, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1651.png", "answer": "x^2/4-y^2=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;O:Origin;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);L:Line;PointOnCurve(RightVertex(C),L);IsPerpendicular(xAxis,L);Intersection(L,OneOf(Asymptote(C)))=A;RightFocus(C)=Center(G);Radius(G)=4;PointOnCurve(A,G);PointOnCurve(O,G)", "coords": {} }, { "index": 1652, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1653, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1653.png", "answer": "4/3", "fact_expr": "G: Parabola;Q: Point;P: Point;F: Point;H:Line;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);PointOnCurve(P,H);IsPerpendicular(H,Directrix(G));FootPoint(H,Directrix(G))=Q;AngleOf(Q, P, F) = ApplyUnit(120, degree)", "coords": {} }, { "index": 1654, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1654.png", "answer": "a^2*b^2/(b^2-a^2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;b > a;a > 0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B))", "coords": {} }, { "index": 1655, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1655.png", "answer": "1/2", "fact_expr": "O: Point;Vertex(G)=O;F: Point;Focus(G) =F;G: Parabola;Expression(G) = (y = x^2/2);H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H,G) = {A, B}", "coords": {} }, { "index": 1656, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_1656.png", "answer": "x^2/9-y^2/16=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G)=Focus(H);Expression(Asymptote(G))=(y=pm*(4/3)*x)", "coords": {} }, { "index": 1657, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1657.png", "answer": "x^2+y^2=4/3", "fact_expr": "l: Line;Gamma: Ellipse;H: Circle;O: Origin;P: Point;Q: Point;Expression(Gamma)=(x^2/4+y^2/2=1);Intersection(l,Gamma) = {P, Q};IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q));Center(H)=O;IsTangent(l,H)", "coords": {} }, { "index": 1658, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1658.png", "answer": "(1/2,2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);Focus(G) = F;F: Point;A: Point;Coordinate(A) = (1, 2);P: Point;PointOnCurve(P, G) = True;WhenMin(Perimeter(TriangleOf(P,A,F))) = True", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 1659, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1659.png", "answer": "pm*1", "fact_expr": "G: Hyperbola;H: Line;m: Number;A: Point;B: Point;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = m + x);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = sqrt(2)*(8/3)", "coords": {} }, { "index": 1660, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1661, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/test/hyperbola/problem_1661.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/3 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = sqrt(2);G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 1662, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1662.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (n*y^2 + x^2 = 1);n: Real;H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 1663, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1664, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1664.png", "answer": "3", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*2*sqrt(2)*x)", "coords": {} }, { "index": 1665, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1665.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Length(ImageinaryAxis(G))=2;FocalLength(G) = 2*sqrt(3)", "coords": {} }, { "index": 1666, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1666.png", "answer": "x^2/4+y^2=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 2 = 0);Coordinate(MidPoint(LineSegmentOf(A, B))) = (m, 1/2);Intersection(G, H) = {A, B};Abs(LineSegmentOf(A, B)) = sqrt(5);m:Number", "coords": {} }, { "index": 1667, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1668, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1668.png", "answer": "sqrt(2)-1", "fact_expr": "C: Parabola;p: Number;lambda:Number;T: Point;P: Point;F: Point;Q: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Intersection(Directrix(C),xAxis)=T;PointOnCurve(P,C);PointOnCurve(Q,C);IsTangent(LineOf(T,P),C);VectorOf(T, P) = lambda*VectorOf(F, Q);lambda>0", "coords": {} }, { "index": 1669, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1669.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Ellipse;Expression(H) = (x^2/8 + y^2/4 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 1670, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1671, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.9970735311508179, "direction": "right" }, "optimization": { "final_loss": 2.85904548036342e-06, "converged": true, "iterations": 88 }, "output_path": "results/test/parabola/problem_1671.png", "answer": "4\nx=-2", "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, -4);PointOnCurve(H, G)", "coords": { "H": [ 2.0, -4.0 ] } }, { "index": 1672, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1673, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1674, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1674.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Point;m: Number;Coordinate(H) = (m, 3);PointOnCurve(H, G);Distance(H, Focus(G)) = 5", "coords": {} }, { "index": 1675, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1675.png", "answer": "(4*x^2-4*y^2/15=1)&(x<=-1/2)", "fact_expr": "M: Circle;C1:Circle;C2:Circle;Expression(C1)=((x+2)^2+y^2=1);Expression(C2)=((x-2)^2+y^2=4);IsOutTangent(M,C1);IsOutTangent(M,C2)", "coords": {} }, { "index": 1676, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1676.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;O: Origin;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Inclination(H)=(3/4)*pi;Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 1677, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1678, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1678.png", "answer": "[1/2, 1)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b + x^2/a = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;K: Point;PointOnCurve(K, C);d: Number;Max(Distance(K, F)) = d;P: Point;PointOnCurve(P, RightDirectrix(C));LineSegmentOf(P, F) = d", "coords": {} }, { "index": 1679, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1679.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G) = True;F1: Point;OneOf(Focus(G)) = F1;Distance(P, F1) = 2;M: Point;MidPoint(LineSegmentOf(P, F1)) = M;O: Origin", "coords": {} }, { "index": 1680, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1680.png", "answer": "{5/3, 5/4}", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;Expression(OneOf(Asymptote(G))) = (y = (4/3)*x);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 1681, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1681.png", "answer": "(1, 2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1682, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1683, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1683.png", "answer": "5*sqrt(3)", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;N: Point;F: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(F,l);Intersection(l,C)={M,N};VectorOf(M, F) = 4*VectorOf(F, N);Area(TriangleOf(O,M,N))=sqrt(3)*Abs(LineSegmentOf(M,N))", "coords": {} }, { "index": 1684, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1684.png", "answer": "y=-1", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4)", "coords": {} }, { "index": 1685, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1685.png", "answer": "-12", "fact_expr": "l: Line;G: Ellipse;B: Point;A: Point;M: Point;Expression(G) = (x^2/16 + y^2/8 = 1);Coordinate(M) = (1, 1);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = M;k:Number;Slope(l)=k", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 1686, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1686.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;L: Line;F2: Point;P: Point;Q: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,L);Intersection(L, G) = {P, Q};TangentPoint(InscribedCircle(TriangleOf(P,F2,Q)),LineSegmentOf(P,F2))=MidPoint(LineSegmentOf(P,F2));TangentPoint(InscribedCircle(TriangleOf(P,F2,Q)),LineSegmentOf(P,Q))=F1", "coords": {} }, { "index": 1687, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1687.png", "answer": "x^2/4+y^2/2=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;Coordinate(F) = (sqrt(2), 0);RightFocus(C) = F;G: Line;PointOnCurve(F, G);IsPerpendicular(G, xAxis);Length(InterceptChord(G, C)) = 2", "coords": { "F": [ 1.4142135623730951, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1688, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1688.png", "answer": "[3/8,3/4]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;P: Point;PointOnCurve(P, C) = True;Range(Slope(LineOf(P, A1))) = [-2, -1]", "coords": {} }, { "index": 1689, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1689.png", "answer": "sqrt(2) + 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;O: Origin;Q: Point;F: Point;x0: Number;y0: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(Q) = (x0, y0);RightFocus(C) = F;Quadrant(P) = 1;PointOnCurve(P, C);a*y0 + b*x0 = 0;PerpendicularBisector(LineSegmentOf(O, P)) = LineSegmentOf(Q, F);PerpendicularBisector(LineSegmentOf(Q, F)) = LineSegmentOf(O, P)", "coords": {} }, { "index": 1690, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1690.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;a>0;Expression(Asymptote(G)) = (3*x + pm*2*y = 0)", "coords": {} }, { "index": 1691, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1691.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;PointOnCurve(F, l) = True;Slope(l) = -1;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 1692, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1692.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 1693, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1694, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1694.png", "answer": "(x-1)^2+y^2=4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Circle;Focus(G) = Center(H);Radius(H) = 2", "coords": {} }, { "index": 1695, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1695.png", "answer": "6", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;O: Origin;Coordinate(F) = (1, 0);Vertex(C) = O;Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B)))= (2,m);IsChordOf(LineSegmentOf(A, B),C);m:Number", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 1696, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1697, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1697.png", "answer": "19/2", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2);P: Point;PointOnCurve(P, G);M: Point;Projection(P, xAxis) = M;A: Point;Coordinate(A) = (6, 17/2)", "coords": { "A": [ 6.0, 8.5 ] } }, { "index": 1698, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1698.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;M: Point;PointOnCurve(M,LeftPart(G)) = True;L: Line;PointOnCurve(M,L) = True;IsParallel(L,xAxis) = True;Intersection(L,Asymptote(G)) = {P,Q};P: Point;Q: Point;Abs(LineSegmentOf(M, Q))*Abs(LineSegmentOf(P, M)) = 4", "coords": {} }, { "index": 1699, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "y", "x_coef": 5.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1699.png", "answer": "10", "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/5 + y^2/9 = 1);PointOnCurve(M, C);Focus(C) = {F1, F2}", "coords": {} }, { "index": 1700, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1701, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1701.png", "answer": "(sqrt(6)-sqrt(2))/4", "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (-2, 1);Coordinate(B) = (2, 1);PointOnCurve(P, G)", "coords": { "A": [ -2.0, 1.0 ], "B": [ 2.0, 1.0 ] } }, { "index": 1702, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1702.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 2*a*b;Eccentricity(G) = e", "coords": {} }, { "index": 1703, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1704, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1705, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1705.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z1: Line;Z2: Line;OneOf(Asymptote(C)) = Z1;OneOf(Asymptote(C)) = Z2;Negation(Z1=Z2);IsPerpendicular(Z1, Z2);Distance(Vertex(C), Z1) = 1", "coords": {} }, { "index": 1706, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1706.png", "answer": "x^2/4-y^2/12=1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;FocalLength(C) = 8;M: Point;Coordinate(M) = (1, sqrt(3));PointOnCurve(M, Asymptote(C))", "coords": { "M": [ 1.0, 1.7320508075688772 ] } }, { "index": 1707, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1707.png", "answer": "sqrt(3)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;P: Point;Negation(P=A);Negation(P=B);PointOnCurve(P, C);m: Number;n: Number;Slope(LineOf(A, P)) = m;Slope(LineOf(B, P)) = n;WhenMin((a/b)*(3 - 2/(3*m*n)) + 2/(m*n) + 3*(ln(Abs(m)) + ln(Abs(n))))", "coords": {} }, { "index": 1708, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1708.png", "answer": "3", "fact_expr": "E: Hyperbola;Expression(E) = (x^2/4 - y^2/b^2 = 1);Eccentricity(E) = 2;b: Number;b>0;P: Point;PointOnCurve(P, E) = True;A1: Line;A2: Line;Asymptote(E) = {A1, A2}", "coords": {} }, { "index": 1709, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1710, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1710.png", "answer": "25", "fact_expr": "G: Ellipse;m: Number;P: Point;F1:Point;F2:Point;Expression(G) = (y^2/16 + x^2/m = 1);Distance(P, F1) = 3;Distance(P, F2) = 7;PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)", "coords": {} }, { "index": 1711, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1711.png", "answer": "x=-1\n{(4,4),(4,-4)}", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 1712, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1713, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1714, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1714.png", "answer": "{x+3*y-2=0,x-y+2=0}", "fact_expr": "m: Line;G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (x^2 = y);Coordinate(P) = (-1, 1);PointOnCurve(P, m);Intersection(m, xAxis) = A;PointOnCurve(B,G);IsPerpendicular(LineSegmentOf(P,A),LineSegmentOf(P,B));IsTangent(LineSegmentOf(A, B), G)", "coords": { "P": [ -1.0, 1.0 ] } }, { "index": 1715, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1716, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1717, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1717.png", "answer": "sqrt(13)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 6*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;Intersection(l, C) = A;B: Point;Intersection(l, Directrix(C)) = B;VectorOf(F, B) = 3*VectorOf(F, A);O: Origin", "coords": {} }, { "index": 1718, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1718.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;UpperVertex(G) = A;Intersection(PerpendicularBisector(LineSegmentOf(A, F2)), G) = B;PointOnCurve(F1,LineSegmentOf(A,B))", "coords": {} }, { "index": 1719, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1720, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1720.png", "answer": "x^2/4-y^2/5=1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);Focus(G) = Focus(C);Expression(OneOf(Asymptote(C))) = (y = x*(sqrt(5)/2))", "coords": {} }, { "index": 1721, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1721.png", "answer": "10", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/3 = 1);Coordinate(A) = (0, 2*sqrt(3));RightFocus(C) = F;PointOnCurve(P, LeftPart(C))", "coords": { "A": [ 0.0, 3.4641016151377544 ] } }, { "index": 1722, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1723, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1724, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1724.png", "answer": "x^2/10+y^2/4=1", "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(G) = (x^2/5 - y^2 = 1);Coordinate(I) = (0, -2);Focus(G)=Focus(H);PointOnCurve(I, H)", "coords": { "I": [ 0.0, -2.0 ] } }, { "index": 1725, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1725.png", "answer": "2*pi/3\n2*sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1726, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1726.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 1727, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/test/hyperbola/problem_1727.png", "answer": "pm*1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;H: Line;m: Number;B: Point;A: Point;a>0;b>0;a=1;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = 5);Expression(H) = (m + x - y = 0);Eccentricity(C) = sqrt(3);Intersection(H, C) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A,B)),G);Negation(A=B)", "coords": {} }, { "index": 1728, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1728.png", "answer": "y = 2", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8)", "coords": {} }, { "index": 1729, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1729.png", "answer": "x^2/16+y^2/9=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MajorAxis(C)) = 8;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 1730, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1730.png", "answer": "", "fact_expr": "l: Line;G: Parabola;F: Point;M: Point;N: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = A;PointOnCurve(A, l);Intersection(l, G) = {M, N};A:Point", "coords": {} }, { "index": 1731, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1731.png", "answer": "(sqrt(5)+1)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A,G);PointOnCurve(B,G);c:Number;IsPerpendicular(LineSegmentOf(O,A),LineSegmentOf(O,B));Distance(O, LineOf(A,B)) = c", "coords": {} }, { "index": 1732, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1732.png", "answer": "(0, pm*sqrt(2))", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/4 = 1)", "coords": {} }, { "index": 1733, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 6.324555320336759, "major_axis": "x", "x_coef": 49.0, "y_coef": 40.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1733.png", "answer": "sqrt(15)", "fact_expr": "G: Ellipse;A: Point;M: Point;P: Point;Expression(G) = (x^2/49 + y^2/40 = 1);PointOnCurve(P,G);Coordinate(A) = (3, 0);Abs(VectorOf(A, M)) = 1;DotProduct(VectorOf(P, M), VectorOf(A, M)) = 0", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 1734, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.825101613998413, "b": 2.34489402930029, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.7647765222287931e-06, "converged": true, "iterations": 118 }, "output_path": "results/test/ellipse/problem_1734.png", "answer": "1/2", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (2, 3);Coordinate(OneOf(Focus(C))) = (2,0);PointOnCurve(A, C)", "coords": { "A": [ 2.0, 3.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 1735, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1735.png", "answer": "3/8", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;l: Line;PointOnCurve(F, l) = True;Expression(l) = (y = sqrt(3)*(-2*a + x));A: Point;Intersection(l,RightPart(C)) = A;B: Point;Intersection(l,yAxis) = B;Area(TriangleOf(O, B, F)) = 8*sqrt(3);O: Origin", "coords": {} }, { "index": 1736, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1736.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;C: Point;D: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2, 0);Focus(G) = F;PointOnCurve(P, H);Intersection(H, G) = {A, B};Intersection(LineOf(A, F), G) = C;Intersection(LineOf(B, F), G) = D;Slope(LineOf(A, B)) = k1;Slope(LineOf(C, D)) = k2;k1: Number;k2: Number", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 1737, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1738, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1738.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);l1: Line;Expression(l1) = (4*x - 3*y - 7 = 0);l2: Line;Expression(l2) = (y = -1)", "coords": {} }, { "index": 1739, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1739.png", "answer": "(pm*2, 0)\ny = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)", "coords": {} }, { "index": 1740, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1740.png", "answer": "(0, 2]+[8, +oo)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/m = 1);m: Number;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1741, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1741.png", "answer": "5", "fact_expr": "M: Point;PointOnCurve(M, G) = True;G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;PointOnCurve(A, H) = True;H: Circle;Expression(H) = ((x - 5)^2 + (y + 1)^2 = 1)", "coords": {} }, { "index": 1742, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1742.png", "answer": "2*sqrt(2)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;Intersection(Directrix(G), xAxis) = P;H: Line;PointOnCurve(P, H);k: Number;Slope(H) = k;k>0;A: Point;B: Point;Intersection(H, G) = {A, B};F: Point;Focus(G) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 1743, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1743.png", "answer": "-b^2/a^2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;N: Point;Endpoint(RealAxis(G)) = {M, N};Q: Point;PointOnCurve(Q, G);Negation(Q=M);Negation(Q=N);alpha: Number;AngleOf(Q, M, N) = alpha;beta: Number;AngleOf(Q, N, M) = beta", "coords": {} }, { "index": 1744, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1744.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;O: Origin;A: Point;F2: Point;F1: Point;a: Number;b: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Quadrant(A) = 1;PointOnCurve(A, G);DotProduct(VectorOf(O, A), VectorOf(O, F2)) = Abs(VectorOf(O, F2))^2;Expression(LineOf(O, A)) = (y = x*(sqrt(2)/2))", "coords": {} }, { "index": 1745, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 2.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1745.png", "answer": "16", "fact_expr": "C1: Circle;Expression(C1) = ((x - 2)^2 + (y - 2)^2 = 4);C2: Circle;Expression(C2) = ((x + 2)^2 + (y + 1)^2 = 2);A: Point;B: Point;P: Point;PointOnCurve(P, C1);IsChordOf(LineSegmentOf(A, B), C2);Abs(LineSegmentOf(A, B)) = 2", "coords": {} }, { "index": 1746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1746.png", "answer": "1/2", "fact_expr": "C: Hyperbola;O: Origin;N: Point;M: Point;Expression(C) = (x^2 - y^2 = 1);PointOnCurve(M, C);L:Line;PointOnCurve(M, L);IsPerpendicular(L, OneOf(Asymptote(C)));FootPoint(L, OneOf(Asymptote(C)))=N", "coords": {} }, { "index": 1747, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1748, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1749, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1750, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 8.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1750.png", "answer": "26", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/36 - y^2/64 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F2)) = 14", "coords": {} }, { "index": 1751, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1751.png", "answer": "y = pm*(3/2)*x", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;H: Circle;F: Point;A: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2);LeftFocus(G) = F;PointOnCurve(F, l);TangentPoint(l, H) = A;Intersection(l, RightPart(G)) = B;VectorOf(F, A) = (1/3)*VectorOf(F, B)", "coords": {} }, { "index": 1752, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1752.png", "answer": "1", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/3 - y^2 = 1);F: Point;OneOf(Focus(C)) = F", "coords": {} }, { "index": 1753, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1753.png", "answer": "3", "fact_expr": "F: Point;Focus(C) = F;C: Parabola;Expression(C) = (x^2 = 16*y);l: Line;Expression(l) = (y = -1);A: Point;PointOnCurve(A, C) = True;PointOnCurve(A,LineSegmentOf(A, P)) = True;IsPerpendicular(LineSegmentOf(A,P),l) = True;FootPoint(LineSegmentOf(A,P),l) = P;P: Point", "coords": {} }, { "index": 1754, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1755, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1756, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1756.png", "answer": "-1", "fact_expr": "C: Parabola;p: Number;A: Point;F: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 4);PointOnCurve(A, Directrix(C));Focus(C) = F", "coords": { "A": [ -2.0, 4.0 ] } }, { "index": 1757, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1758, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6277244091033936, "b": 1.818325400352478, "orientation": "horizontal" }, "optimization": { "final_loss": 6.648646149187698e-07, "converged": true, "iterations": 99 }, "output_path": "results/test/hyperbola/problem_1758.png", "answer": "y=pm*sqrt(5)*x/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Eccentricity(G) = 3/2", "coords": {} }, { "index": 1759, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1760, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1760.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;P: Point;F: Point;p>0;Expression(G)=(y^2 = 2*p*x);Coordinate(P)=(6, y1);PointOnCurve(P,G);Focus(G)=F;Distance(P,F) = 8;y1:Number", "coords": {} }, { "index": 1761, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1761.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Hyperbola;A: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(A) = (0, 1)", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 1762, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1762.png", "answer": "sqrt(3) + 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));AngleOf(P, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 1763, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1763.png", "answer": "5", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*(x - 1));Coordinate(MidPoint(LineSegmentOf(A, B))) = (3/2, 1);Intersection(H, G) = {A, B};k:Number", "coords": {} }, { "index": 1764, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1764.png", "answer": "2/3", "fact_expr": "F: Point;A: Point;LeftFocus(Gamma) = F;LowerVertex(Gamma) = A;Gamma: Ellipse;Expression(Gamma) = (x^2/2 + y^2 = 1);M: Point;x0: Number;y0: Number;Coordinate(M) = (x0, y0);Quadrant(M) = 1;PointOnCurve(M, Gamma);N: Point;Coordinate(N) = (0, 1/y0);H: Point;PointOnCurve(H, LineSegmentOf(F, M));Abs(VectorOf(F, H)) = Abs(VectorOf(A, H));DotProduct(VectorOf(F, H), VectorOf(N, H)) = 0", "coords": {} }, { "index": 1765, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1765.png", "answer": "2*x+y-3=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/4 = 1);P: Point;Coordinate(P) = (1, 1);H: LineSegment;PointOnCurve(P, H);IsChordOf(H, G);MidPoint(H) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 1766, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1766.png", "answer": "2", "fact_expr": "G: Parabola;l1: Line;l2: Line;Expression(G) = (y = x^2/4);Expression(l1) = (3*x - 4*y - 6 = 0);Expression(l2) = (y = -1);P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 1767, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1767.png", "answer": "3/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: Line;PointOnCurve(F, L);Intersection(L, G) = {P, Q};P: Point;Q: Point;Length(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 1768, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1768.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*x - k);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 1769, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 9.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1769.png", "answer": "7/2", "fact_expr": "G: Ellipse;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/7 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);AngleOf(A, F1, F2) = ApplyUnit(45, degree)", "coords": {} }, { "index": 1770, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1770.png", "answer": "sqrt(3)/3", "fact_expr": "l: Line;G: Hyperbola;M: Point;N: Point;F2: Point;F1: Point;Expression(G) = (x^2/3 - y^2 = 1);RightFocus(G) = F2;LeftFocus(G)=F1;PointOnCurve(F1, l);Inclination(l) = ApplyUnit(60, degree);Intersection(l, G) = {M, N}", "coords": {} }, { "index": 1771, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1772, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1772.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;N: Point;Coordinate(N) = (4, 0);l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};G: Circle;IsDiameter(LineSegmentOf(N, F), G);M: Point;Intersection(l, G) = M;Negation(M=F);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "N": [ 4.0, 0.0 ] } }, { "index": 1773, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1773.png", "answer": "1+sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F2: Point;RightFocus(G) = F2;P: Point;Q: Point;IsChordOf(LineSegmentOf(P,Q),G) = True;PointOnCurve(F2,LineSegmentOf(P,Q)) = True;IsPerpendicular(LineSegmentOf(P,Q),RealAxis(G)) = True;F1: Point;LeftFocus(G) = F1;AngleOf(P, F1, Q) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1774, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1774.png", "answer": "9/2", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a1^2 + y^2/b1^2 = 1);a1: Number;b1: Number;a1 > 0;b1 > 0;C2: Hyperbola;Expression(C2) = (x^2/a2^2 - y^2/b2^2 = 1);a2: Number;b2: Number;a2 > 0;b2 > 0;F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};OneOf(Intersection(C1, C2)) = P;P: Point;e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True", "coords": {} }, { "index": 1775, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1775.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F: Point;O: Origin;M: Point;x1:Number;y1:Number;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(P) = (x1, y1);Coordinate(F) = (3, 0);PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 2;MidPoint(LineSegmentOf(P, F)) = M", "coords": { "F": [ 3.0, 0.0 ] } }, { "index": 1776, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1776.png", "answer": "2", "fact_expr": "G: Parabola;l1: Line;l2: Line;P: Point;Expression(G) = (y = x^2/4);PointOnCurve(P, G);Expression(l1) = (4*x - 3*y = 7);Expression(l2) = (y = -1)", "coords": {} }, { "index": 1777, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1777.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F2, G);Slope(G) = b/a;P: Point;Intersection(G, C) = P;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 1778, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1778.png", "answer": "pi/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;Intersection(Directrix(C), xAxis) = M;P: Point;Q: Point;l1: Line;l2: Line;TangentOfPoint(M, C) = {l1, l2};TangentPoint(l1, C) = P;TangentPoint(l2, C) = Q", "coords": {} }, { "index": 1779, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1779.png", "answer": "sqrt(2)", "fact_expr": "Gamma: Hyperbola;l: Line;a: Number;b: Number;a>0;b>0;A: Point;B: Point;Expression(l) = (x - sqrt(3)*y = 0);Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);Intersection(l, Gamma) = {A, B};C: Point;PointOnCurve(A, LineOf(A, C));IsPerpendicular(LineOf(A, C), l);Intersection(LineOf(A, C), Gamma) = C;AngleOf(A, B, C) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1780, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1780.png", "answer": "1/4", "fact_expr": "l: Line;G: Parabola;M: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (2, 0);Negation(IsPerpendicular(l,xAxis));PointOnCurve(M, l);Intersection(l, G) = {A, B}", "coords": { "M": [ 2.0, 0.0 ] } }, { "index": 1781, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1781.png", "answer": "10", "fact_expr": "G: Ellipse;A: Point;F: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);RightFocus(G) = F;Coordinate(A) = (0, sqrt(6))", "coords": { "A": [ 0.0, 2.449489742783178 ] } }, { "index": 1782, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1782.png", "answer": "16", "fact_expr": "M: Parabola;Expression(M) = (y^2 = 4*x);F: Point;Focus(M) = F;A: Point;B: Point;C: Point;D: Point;PointOnCurve(F, LineSegmentOf(A, B));PointOnCurve(F, LineSegmentOf(C, D));IsChordOf(LineSegmentOf(A, B), M);IsChordOf(LineSegmentOf(C, D), M);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D));Intersection(LineSegmentOf(A, B), M) = {A, B};Intersection(LineSegmentOf(C, D), M) = {C, D}", "coords": {} }, { "index": 1783, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1783.png", "answer": "8", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l,G) ={A,B}", "coords": {} }, { "index": 1784, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1784.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;Q: Point;F: Point;P: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,l);Intersection(l,LeftPart(C))=P;Intersection(l,RightPart(C))=Q;Abs(LineSegmentOf(Q,F))-Abs(LineSegmentOf(P,F))=Length(ImageinaryAxis(C))", "coords": {} }, { "index": 1785, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1785.png", "answer": "3", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;Q: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C)=F;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);PointOnCurve(Q, xAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P));Abs(LineSegmentOf(F, Q)) = 6", "coords": {} }, { "index": 1786, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1787, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1787.png", "answer": "1", "fact_expr": "C: Parabola;M: Point;O:Origin;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Quadrant(P)=1;MidPoint(LineSegmentOf(P, F)) = M;Vertex(C)=O", "coords": {} }, { "index": 1788, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1788.png", "answer": "4*sqrt(13)", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (y^2 = -16*x);Focus(G) = F;PointOnCurve(P, Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 8", "coords": {} }, { "index": 1789, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1790, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1791, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1792, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1792.png", "answer": "sqrt(3)/3", "fact_expr": "l: Line;E: Ellipse;b: Number;a: Number;P: Point;F2: Point;Q: Point;F1: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F2,l);Intersection(l, E) = {P, Q};Abs(LineSegmentOf(P, F2)) = 2*Abs(LineSegmentOf(Q,F2));Abs(LineSegmentOf(Q,F1))=3*Abs(LineSegmentOf(Q,F2))", "coords": {} }, { "index": 1793, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1793.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;D: Point;C: Point;A: Point;M: Point;B: Point;F: Point;k2: Number;k1: Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (4, 0);Focus(G) = F;PointOnCurve(F, H);Slope(H) = k1;Intersection(H, G) = {A, B};Intersection(OverlappingLine(LineSegmentOf(A, M)), G) = C;Intersection(OverlappingLine(LineSegmentOf(B, M)), G) = D;Slope(LineOf(C, D)) = k2", "coords": { "M": [ 4.0, 0.0 ] } }, { "index": 1794, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1794.png", "answer": "4+pm*2*sqrt(3)", "fact_expr": "C: Parabola;P: Point;F: Point;H: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = H;PointOnCurve(P, C);Abs(LineSegmentOf(P, H)) = (sqrt(5)/2)*Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 1795, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1795.png", "answer": "(sqrt(2) + 1, +oo)", "fact_expr": "C: Hyperbola;a: Number;b: Number;N: Point;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;H: Line;PointOnCurve(F2,H);IsPerpendicular(H,xAxis);Intersection(H, C) = {M, N};DotProduct(VectorOf(N, F1), VectorOf(M, F1)) < 0", "coords": {} }, { "index": 1796, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1796.png", "answer": "y^2=16*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;M: Point;PointOnCurve(M, G) = True;Abs(LineSegmentOf(M, F)) = 3*Abs(LineSegmentOf(O, F));Area(TriangleOf(M, F, O)) = 16*sqrt(2)", "coords": {} }, { "index": 1797, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/test/hyperbola/problem_1797.png", "answer": "1", "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Eccentricity(C)=sqrt(5);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(F2, P));Area(TriangleOf(P, F1, F2)) = 4", "coords": {} }, { "index": 1798, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1798.png", "answer": "sqrt(5)", "fact_expr": "C1: Hyperbola;a: Number;C2: Hyperbola;b: Number;a>0;b>0;Expression(C1) = (y^2/a^2 - x^2/b^2 = 1);Expression(C2) = (x^2/4 - y^2 = 1);Asymptote(C1) = Asymptote(C2)", "coords": {} }, { "index": 1799, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1799.png", "answer": "(11/10,0)", "fact_expr": "A: Point;B: Point;A1: Point;F: Point;G:Curve;C:Point;Expression(G)=(x^2/16-y^2/9=1);IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));RightFocus(G)=F;PointOnCurve(A,LineSegmentOf(A,A1));IsPerpendicular(RightDirectrix(G),LineSegmentOf(A,A1));FootPoint(RightDirectrix(G),LineSegmentOf(A,A1))=A1;Intersection(LineSegmentOf(B,A1),xAxis)=C", "coords": {} }, { "index": 1800, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.9876673221588135, "b": 1.5893483050269506, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.171300880237459e-05, "converged": true, "iterations": 68 }, "output_path": "results/test/ellipse/problem_1800.png", "answer": "16", "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/25 + y^2/k = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 3/5", "coords": {} }, { "index": 1801, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1802, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1802.png", "answer": "pm*sqrt(3)", "fact_expr": "l: Line;G: Ellipse;A: Point;F: Point;B: Point;k: Number;Expression(G) = (x^2/9 + y^2/5 = 1);RightFocus(G) = F;PointOnCurve(F, l);Slope(l) = k;Intersection(l, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 1803, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1803.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;Expression(G) = (y^2 = -2*p*x);Expression(H) = (y^2 + (x + 1)^2 = 1);Center(H) = Focus(G)", "coords": {} }, { "index": 1804, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1532304286956787, "b": 2.5773532390594482, "orientation": "horizontal" }, "optimization": { "final_loss": 8.412490387854632e-06, "converged": true, "iterations": 114 }, "output_path": "results/test/hyperbola/problem_1804.png", "answer": "sqrt(6)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(5)*x)", "coords": {} }, { "index": 1805, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1805.png", "answer": "[1,3)+(16/3,16]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/4 + x^2/a = 1);a: Number;e: Number;Eccentricity(G) = e;In(e,(1/2, sqrt(3)/2])", "coords": {} }, { "index": 1806, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1807, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1807.png", "answer": "x^2/2-y^2/4=1", "fact_expr": "G: Hyperbola;H: Point;C:Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);Coordinate(H) = (2, 2);Asymptote(G) = Asymptote(C);PointOnCurve(H, C)", "coords": { "H": [ 2.0, 2.0 ] } }, { "index": 1808, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1808.png", "answer": "", "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;Expression(G) = (m*x^2 + n*y^2 = 1);Expression(H) = (y = x^2/8);OneOf(Focus(G)) = Focus(H);Eccentricity(G)=2", "coords": {} }, { "index": 1809, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1809.png", "answer": "2*sqrt(13)", "fact_expr": "C: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);PointOnCurve(P, Directrix(C));Abs(LineSegmentOf(A, F)) = 5", "coords": {} }, { "index": 1810, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1810.png", "answer": "[sqrt(3)/2,1)", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(P,C);Quadrant(P)=1;DotProduct(VectorOf(F, O), VectorOf(F, P)) = Abs(LineSegmentOf(P, F))^2", "coords": {} }, { "index": 1811, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1811.png", "answer": "y^2 = -16*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -2*p*x);p: Number;p>0;H: Circle;Expression(H) = (-4*x + x^2 + y^2 = 0);IsTangent(H, Directrix(G)) = True", "coords": {} }, { "index": 1812, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1812.png", "answer": "4*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C), xAxis) = M;M: Point;F: Point;Focus(C) = F;L: Line;Inclination(L) = ApplyUnit(60,degree);PointOnCurve(F,L) = True;Intersection(L,C) = {A,B};A: Point;B: Point", "coords": {} }, { "index": 1813, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1813.png", "answer": "y^2 = 2*x - 2", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(B) = (2, 0);MidPoint(LineSegmentOf(A, B)) = P;PointOnCurve(A, G)", "coords": { "B": [ 2.0, 0.0 ] } }, { "index": 1814, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1814.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/8 = 1);Expression(H) = (x + 2*y + 7 = 0)", "coords": {} }, { "index": 1815, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1816, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1817, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1817.png", "answer": "pm*3", "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;A: Point;O: Origin;Expression(C) = (y^2 = 2*p*x);PointOnCurve(A, C);Focus(C) = F;Distance(A, F) = 4;PointOnCurve(M, Directrix(C));VectorOf(M, F) = 3*VectorOf(F, A)", "coords": {} }, { "index": 1818, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1818.png", "answer": "[-1,0)", "fact_expr": "x0: Number;x0 < 0;M: Point;Coordinate(M) = (x0, 0);O: Origin;C: Parabola;Expression(C) = (y^2 = 4*x);N: Point;PointOnCurve(N, C);AngleOf(O, M, N) = ApplyUnit(45, degree)", "coords": {} }, { "index": 1819, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1819.png", "answer": "7", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P,F1)), yAxis);Abs(LineSegmentOf(P, F1)) = lambda*Abs(LineSegmentOf(P, F2));lambda:Number", "coords": {} }, { "index": 1820, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1820.png", "answer": "sqrt(5)/2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;O: Origin;Expression(G) = (y^2 = 2*x);Coordinate(A) = (-1/2, 0);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(A, P)) = sqrt(2)*Abs(LineSegmentOf(P, F))", "coords": { "A": [ -0.5, 0.0 ] } }, { "index": 1821, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1821.png", "answer": "20", "fact_expr": "C: Hyperbola;b: Number;G: Circle;H: Circle;F2: Point;P: Point;F1: Point;A1: Point;A2: Point;b>0;Expression(C) = (x^2/5 - y^2/b^2 = 1);LeftVertex(C) = A1;RightVertex(C) = A2;LeftFocus(C)=F1;RightFocus(C)=F2;IsDiameter(LineSegmentOf(F1,F2),G);OneOf(Intersection(G,LeftPart(C)))=P;IsDiameter(LineSegmentOf(A1,A2),H);IsTangent(H,LineOf(P,F2))", "coords": {} }, { "index": 1822, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1823, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1823.png", "answer": "sqrt(2)-1", "fact_expr": "N: Parabola;p: Number;M: Ellipse;a: Number;b: Number;P: Point;F1: Point;F2: Point;Expression(N) = (y^2 = 2*(p*x));a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;Focus(N) = F2;OneOf(Intersection(M,N))=P;Abs(LineSegmentOf(P, F1)) = sqrt(2)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1824, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1824.png", "answer": "x^2/4-y^2/3=1", "fact_expr": "G: Ellipse;A: Point;M: Point;N: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftVertex(G) = A;RightVertex(G) = B;PointOnCurve(M,G);PointOnCurve(N,G);IsPerpendicular(LineSegmentOf(M,N),xAxis)", "coords": {} }, { "index": 1825, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1825.png", "answer": "-11", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = 2*x - 2);Intersection(H, G) = {A, B};Focus(G) = F", "coords": {} }, { "index": 1826, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1826.png", "answer": "{(-2/3, sqrt(2)/3), (-2/3, -sqrt(2)/3)}", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;Z: Line;PointOnCurve(O, Z);IsPerpendicular(Z, LineOf(A, B));H: Point;FootPoint(Z, LineOf(A, B)) = H", "coords": {} }, { "index": 1827, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1827.png", "answer": "1", "fact_expr": "l: Line;C: Curve;O: Origin;P: Point;Q: Point;Expression(C) = (y = x^2/2);PointOnCurve(P, C);TangentOnPoint(P, C) = l;Z: Line;PointOnCurve(P, Z);IsPerpendicular(Z, l);Intersection(Z, C) = Q;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q))", "coords": {} }, { "index": 1828, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 3.9731175899505615, "direction": "right" }, "optimization": { "final_loss": 0.00018243117665406317, "converged": true, "iterations": 127 }, "output_path": "results/test/parabola/problem_1828.png", "answer": "5", "fact_expr": "G: Parabola;a: Number;P: Point;F: Point;Expression(G) = (y^2 = a*x);Coordinate(P) = (1, 4);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = Abs(LineSegmentOf(P, F));Negation(a = 0)", "coords": { "P": [ 1.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 1829, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/test/hyperbola/problem_1829.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;F1: Point;F2: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (3, 4);PointOnCurve(A, C);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);PointOnCurve(A, G)", "coords": { "A": [ 3.0, 4.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1830, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1830.png", "answer": "2", "fact_expr": "G: Ellipse;O: Circle;H: Line;m: Number;n: Number;I: Point;L:Line;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(O) = (x^2 + y^2 = 4);Expression(H) = (m*x + n*y = 4);Coordinate(I) = (m, n);NumIntersection(H, O)=0;PointOnCurve(I,L)", "coords": {} }, { "index": 1831, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1832, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1833, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1834, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1834.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Line;P: Point;F1: Point;Q: Point;F2: Point;a > b;b > 0;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {P, Q};AngleOf(F1, P, Q) = ApplyUnit(60, degree);Abs(LineSegmentOf(P,F1))=Abs(LineSegmentOf(P,Q))", "coords": {} }, { "index": 1835, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1835.png", "answer": "5/13", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);P: Point;PointOnCurve(P, G) = True;PointOnCurve(MidPoint(LineSegmentOf(P,F1)),yAxis) = True", "coords": {} }, { "index": 1836, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1836.png", "answer": "pi/6", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 2*sqrt(3)/3;H: Line;Expression(H) = (y = x*(b/a))", "coords": {} }, { "index": 1837, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1838, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1838.png", "answer": "5/7", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;O: Origin;PointOnCurve(O, G);A: Point;B: Point;Intersection(C, G) = {A,B};Abs(LineSegmentOf(A, B)) = 10;Abs(LineSegmentOf(A, F)) = 6;Cos(AngleOf(A, B, F)) = 4/5;e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 1839, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1839.png", "answer": "(1, sqrt(2)]", "fact_expr": "C1:Hyperbola;C2:Hyperbola;a:Number;b:Number;a>0;b>0;m:Number;n:Number;m>0;n>0;e1:Number;e2:Number;Expression(C1)=(x^2/a^2-y^2/b^2=1);Expression(C2)=(y^2/m^2-x^2/n^2=1);Eccentricity(C1)=e1;Eccentricity(C2)=e2;Asymptote(C1)=Asymptote(C2)", "coords": {} }, { "index": 1840, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1840.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);k: Number;Slope(l) = k;P: Point;Q: Point;Intersection(l, C) = {P, Q};M: Point;Intersection(PerpendicularBisector(LineSegmentOf(P, Q)), xAxis) = M;Abs(LineSegmentOf(P, Q)) = 6", "coords": {} }, { "index": 1841, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1841.png", "answer": "", "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 6);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 4.0, 6.0 ] } }, { "index": 1842, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1842.png", "answer": "2*sqrt(5) - 1", "fact_expr": "G: Parabola;H: Line;A: Point;m:Number;n:Number;Expression(G) = (x^2 = 4*y);Expression(H) = (x + 2*y + 8 = 0);PointOnCurve(A, G);Distance(A, xAxis) = m;Distance(A,H)=n", "coords": {} }, { "index": 1843, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1843.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;Slope(G) = sqrt(2);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin;P: Point;MidPoint(LineSegmentOf(A, B)) = P;Slope(LineOf(O, P)) = 2*sqrt(2)", "coords": {} }, { "index": 1844, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1845, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1845.png", "answer": "7", "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;M:Point;Expression(G) = (y^2 = 6*x);PointOnCurve(Focus(G),l);Intersection(l,G)={A,B};MidPoint(LineSegmentOf(A,B))=M;XCoordinate(M)=2", "coords": {} }, { "index": 1846, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1846.png", "answer": "(15*sqrt(41))/41", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (4*x - 5*y + 40 = 0)", "coords": {} }, { "index": 1847, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 9.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1847.png", "answer": "[5, 7]", "fact_expr": "C: Ellipse;F2: Point;P: Point;F1: Point;Q: Point;F: Circle;Expression(C) = (x^2/9 + y^2/6 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Center(F) = F2;Radius(F) = 1;PointOnCurve(P, C);PointOnCurve(Q, F)", "coords": {} }, { "index": 1848, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1849, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1849.png", "answer": "sqrt(5)/5", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);IsPerpendicular(H, xAxis);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;Intersection(LineOf(A, F2), G) = {A, C};VectorOf(A, F2) = 2*VectorOf(F2, C)", "coords": {} }, { "index": 1850, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1850.png", "answer": "", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);H: LineSegment;IsChordOf(H,G) = True;Coordinate(MidPoint(H)) = (1/2, 1/2)", "coords": {} }, { "index": 1851, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1851.png", "answer": "pm*2*sqrt(2)/3", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (-1, 0);PointOnCurve(P, H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(P, B)) = A", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 1852, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1852.png", "answer": "", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/16 + y^2/9 = 1)", "coords": {} }, { "index": 1853, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1854, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1854.png", "answer": "2", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - 4*y^2 = 4);Focus(G) = {F1, F2};PointOnCurve(P, G);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0", "coords": {} }, { "index": 1855, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.24967060983181, "direction": "up" }, "optimization": { "final_loss": 6.555423624376999e-06, "converged": true, "iterations": 116 }, "output_path": "results/test/parabola/problem_1855.png", "answer": "(0, 1/4)", "fact_expr": "G: Parabola;m: Number;H: Point;Expression(G) = (x^2 = m*y);Coordinate(H) = (-1, 1);PointOnCurve(H, G)", "coords": { "H": [ -1.0, 1.0 ] } }, { "index": 1856, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1857, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1858, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1858.png", "answer": "6", "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 8", "coords": {} }, { "index": 1859, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1860, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1860.png", "answer": "4/3", "fact_expr": "C: Parabola;k: Number;k > 0;F: Point;A: Point;B: Point;A1: Point;B1: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;H: Line;PointOnCurve(F, H);Slope(H) = k;Intersection(H, C) = {A, B};l1: Line;l2: Line;PointOnCurve(A, l1);PointOnCurve(B, l2);IsPerpendicular(l1, Directrix(C));IsPerpendicular(l2, Directrix(C));FootPoint(l1, Directrix(C)) = A1;FootPoint(l2, Directrix(C)) = B1;Area(TriangleOf(A, B, B1))/Area(TriangleOf(A, B, A1)) = 4", "coords": {} }, { "index": 1861, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1861.png", "answer": "", "fact_expr": "G: Hyperbola;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 - y^2 = 1);PointOnCurve(RightFocus(G), H);Inclination(H)=ApplyUnit(45, degree) ;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1862, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1862.png", "answer": "(x-3)^2+y^2=5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);H: Circle;Center(H) = RightFocus(G);IsTangent(H, Asymptote(G))", "coords": {} }, { "index": 1863, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1864, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1864.png", "answer": "3*sqrt(2)", "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1);PointOnCurve(P, G)", "coords": {} }, { "index": 1865, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1865.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsPerpendicular(H, xAxis);OneOf(Intersection(H, G)) = P;AngleOf(P, F1, F2) = pi/6", "coords": {} }, { "index": 1866, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1867, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1867.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 6", "coords": {} }, { "index": 1868, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/test/hyperbola/problem_1868.png", "answer": "x^2/5-y^2/20=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = 2*x);H: Parabola;Expression(H) = (y^2 = 20*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 1869, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1870, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1871, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1871.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, Asymptote(C));IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2));Abs(LineSegmentOf(M, F1)) = 2*b + Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 1872, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1872.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;Expression(l) = (x - sqrt(3)*y + 8 + 2*sqrt(3) = 0);P: Point;PointOnCurve(P, l);WhenMax(AngleOf(F1, P, F2))", "coords": {} }, { "index": 1873, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1873.png", "answer": "x^2/9-y^2/3=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C)=(-y^2/b^2 + x^2/a^2=1);Expression(E)=(x^2 + (y-1/2)^2=49/4);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,RightPart(C));IsDiameter(LineSegmentOf(P,F1),E);PointOnCurve(F2,E)", "coords": {} }, { "index": 1874, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1875, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1875.png", "answer": "2*sqrt(2)", "fact_expr": "C: Hyperbola;P:Point;Expression(C) = (x^2/2 - y^2 = 1);PointOnCurve(P, C);l1:Line;l2:Line;Asymptote(C)={l1,l2};Distance(P,l1)=d1;Distance(P,l2)=d2;d1:Number;d2:Number", "coords": {} }, { "index": 1876, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1877, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1877.png", "answer": "1/2", "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;D: Point;lambda:Number;Expression(G) = (y^2 = 2*x);PointOnCurve(Focus(G),LineSegmentOf(A,B));PointOnCurve(Focus(G),LineSegmentOf(C,D));IsChordOf(LineSegmentOf(A,B),G);IsChordOf(LineSegmentOf(C,D),G);IsPerpendicular(LineSegmentOf(A,B),LineSegmentOf(C,D));Abs(LineSegmentOf(A, B)) + Abs(LineSegmentOf(C, D)) = (lambda*Abs(LineSegmentOf(A, B)))*Abs(LineSegmentOf(C, D))", "coords": {} }, { "index": 1878, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1879, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1879.png", "answer": "2\nApplyUnit(120, degree)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1880, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1881, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 2.449489742783178, "major_axis": "x", "x_coef": 10.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1881.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/10 + y^2/6 = 1)", "coords": {} }, { "index": 1882, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1882.png", "answer": "-6", "fact_expr": "M: Point;H: Circle;Expression(H) = ((x - 6)^2 + (y - 4)^2 = 1);PointOnCurve(M, H);P: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F: Point;Coordinate(F) = (-3, 0)", "coords": { "F": [ -3.0, 0.0 ] } }, { "index": 1883, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1883.png", "answer": "9/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G, LeftPart(C)) = {A, B};Abs(LineSegmentOf(A, F2)) = 3;Abs(LineSegmentOf(B, F2)) = 5;Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 1884, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1885, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1886, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1886.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3^2 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 1887, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1887.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;F: Point;B: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (-2, 0);PointOnCurve(M, l);Slope(l) = 2/3;Intersection(l, C) = {A, B};Focus(C)=F;Area(TriangleOf(M,F,B))=2*Area(TriangleOf(M,F,A))", "coords": { "M": [ -2.0, 0.0 ] } }, { "index": 1888, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1888.png", "answer": "(1,3]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, LeftPart(G));Abs(LineSegmentOf(P, F2))^2/Abs(LineSegmentOf(P, F1)) = 8*a", "coords": {} }, { "index": 1889, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1890, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1890.png", "answer": "y^2 = 4*x", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 2*p*x);p: Number;M: Point;Coordinate(M) = (1, m);m: Number;PointOnCurve(M, E) = True;Distance(M, Focus(E)) = 2", "coords": {} }, { "index": 1891, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1891.png", "answer": "[2, 2*sqrt(2)]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);x0^2/2 + y0^2 <= 1", "coords": {} }, { "index": 1892, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1892.png", "answer": "6", "fact_expr": "l: Line;G: Hyperbola;O: Origin;c: Number;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c;Expression(l) = (a*x + b*y = a*b);Distance(O, l) = (1/3)*c + 1", "coords": {} }, { "index": 1893, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1893.png", "answer": "y=-x+1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(G) = F;l: Line;PointOnCurve(F, l) = True;Intersection(l, G) = {P, Q};Q: Point;P: Point;Quadrant(P) = 2;Q1: Point;PointOnCurve(Q, G) = True;PointOnCurve(Q1, G) = True;Coordinate(Q) = (xQ, yQ);xQ: Number;yQ: Number;Coordinate(Q1) = (xQ1, yQ1);xQ1: Number;yQ1: Number;yQ + yQ1 = 0;IsPerpendicular(LineSegmentOf(F, Q1), LineSegmentOf(P, Q)) = True", "coords": {} }, { "index": 1894, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1894.png", "answer": "2", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 2);Expression(H) = (y^2 = 2*(p*x));Directrix(H) = LeftDirectrix(G)", "coords": {} }, { "index": 1895, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1895.png", "answer": "[2, 3]", "fact_expr": "E: Ellipse;F1: Point;P: Point;F2: Point;Expression(E) = (x^2/4 + y^2/3 = 1);Focus(E) = {F1, F2};PointOnCurve(P, E)", "coords": {} }, { "index": 1896, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1896.png", "answer": "x^2/4+3*y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;C: Point;A: Point;O: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (2, 0);Center(G) = O;OneOf(Endpoint(MajorAxis(G)))=A;PointOnCurve(O, LineSegmentOf(B, C));IsChordOf(LineSegmentOf(B, C),G);DotProduct(VectorOf(A, C), VectorOf(B, C)) = 0;Abs(-VectorOf(O, B) + VectorOf(O, C)) = 2*Abs(-VectorOf(B, A) + VectorOf(B, C))", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 1897, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1897.png", "answer": "25/4", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C)=(y^2 = 4*x);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 5", "coords": {} }, { "index": 1898, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1898.png", "answer": "sqrt(2)\n(pm*sqrt(2), 0)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/3 + x^2/m = 1);m: Number;m>0;T:Point;OneOf(Focus(G))=T;Coordinate(T) = (0, 1);PointOnCurve(P, G) ;P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};Area(TriangleOf(P, F1, F2)) = sqrt(2)", "coords": { "T": [ 0.0, 1.0 ] } }, { "index": 1899, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1900, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1900.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (0, 1);PointOnCurve(P, G)", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 1901, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1901.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;J: Line;H: Line;A: Point;P: Point;B: Point;E: Point;F: Point;O: Origin;Expression(G) = (x^2/4 + y^2/2 = 1);Expression(H) = (x = 3);RightVertex(G) = P;PointOnCurve(O, J);Intersection(J, G) = {A, B};Intersection(LineOf(P, A), H) = E;Intersection(LineOf(P, B), H) = F", "coords": {} }, { "index": 1902, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1902.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);XCoordinate(P) = 2", "coords": {} }, { "index": 1903, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1904, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1904.png", "answer": "pm*sqrt(210)/3", "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (x^2/3 - y^2/2 = 1);Length(InterceptChord(l,G))=4;Slope(l)=2", "coords": {} }, { "index": 1905, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1905.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/3 = 1);H: Circle;Expression(H) = (y^2 + (x - 3)^2 = r^2);r: Number;r>0;IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 1906, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1906.png", "answer": "y = pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/4 = 1)", "coords": {} }, { "index": 1907, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1907.png", "answer": "{2*sqrt(3)/3,2}", "fact_expr": "C: Hyperbola;G: Circle;O: Origin;Expression(G) = (y^2 + (x - 2)^2 = 1);Center(C) = O;SymmetryAxis(C)=axis;IsTangent(Asymptote(C),G)", "coords": {} }, { "index": 1908, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1908.png", "answer": "1/3", "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(H,G))=P", "coords": {} }, { "index": 1909, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1909.png", "answer": "2*sqrt(38)/5", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 4);Expression(H) = (y = x - 1/2)", "coords": {} }, { "index": 1910, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1910.png", "answer": "sqrt(17)/3", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G));DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 1/4", "coords": {} }, { "index": 1911, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1911.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L1:Line;L2:Line;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2);L:Line;PointOnCurve(F, L);IsPerpendicular(L1, L);FootPoint(L1, L)=A;Intersection(L,L2)=B;2*VectorOf(A, F) = VectorOf(F, B)", "coords": {} }, { "index": 1912, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1912.png", "answer": "1 + sqrt(2)", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;Q: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};IsChordOf(LineSegmentOf(P, Q), G);PointOnCurve(F1, LineSegmentOf(P, Q));IsPerpendicular(xAxis, LineSegmentOf(P, Q));AngleOf(P, F2, Q) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1913, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1913.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1)", "coords": {} }, { "index": 1914, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1914.png", "answer": "8", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;N: Point;M: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x + 3)^2 = 1);Expression(C) = (y^2 + (x - 3)^2 = 1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, C)", "coords": {} }, { "index": 1915, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1915.png", "answer": "2", "fact_expr": "G: Parabola;l: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (3*x - 4*y + 12 = 0);PointOnCurve(P, G);d1:Number;d2:Number;Distance(P, yAxis) = d1;Distance(P, l) = d2", "coords": {} }, { "index": 1916, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1916.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(M) = (y^2 + (-a + x)^2 = b^2/4);IsTangent(OneOf(Asymptote(C)),M);WhenMax(a^2/(a^2*b^2+1)-(4/49)*a^2)", "coords": {} }, { "index": 1917, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1917.png", "answer": "y^2 = 8*x", "fact_expr": "l: Line;Slope(l) = 2;G: Parabola;Expression(G) = (y^2 = a*x);a>0;a: Number;F: Point;PointOnCurve(F,l) = True;Focus(G) = F;Intersection(l,yAxis) = A;O: Origin;A: Point;Area(TriangleOf(O,A,F)) = 4", "coords": {} }, { "index": 1918, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1918.png", "answer": "y=(pm*3/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 1919, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1920, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1920.png", "answer": "4+2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True", "coords": {} }, { "index": 1921, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1921.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Circle;A: Point;C: Point;B: Point;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = a^2);LeftFocus(G) = F;l1: Line;l2: Line;TangentOfPoint(F, O) = {l1, l2};TangentPoint(l1,O)=A;TangentPoint(l2,O)=B;LeftVertex(G) = C;AngleOf(A, C, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 1922, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1922.png", "answer": "sqrt(17)-1", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C);G: Circle;Expression(G) = (x^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q, G);Distance(P, Directrix(C)) = d;d: Number", "coords": {} }, { "index": 1923, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1923.png", "answer": "(sqrt(5)+1)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;OneOf(Focus(G)) = F;F: Point;OneOf(Endpoint(ImageinaryAxis(G))) = B;B: Point;IsPerpendicular(LineOf(F,B),OneOf(Asymptote(G))) = True", "coords": {} }, { "index": 1924, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1924.png", "answer": "x^2=6*y-9", "fact_expr": "G: Parabola;F: Point;M: Point;P: Point;Expression(G) = (x^2 = 12*y);Focus(G) = F;PointOnCurve(P, G);VectorOf(F, M) = VectorOf(M, P)", "coords": {} }, { "index": 1925, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1925.png", "answer": "x^2/3 - y^2/12 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2 - y^2/4 = 1);Coordinate(H) = (2, 2);Asymptote(G) = Asymptote(C);PointOnCurve(H, C)", "coords": { "H": [ 2.0, 2.0 ] } }, { "index": 1926, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1926.png", "answer": "(1, 2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, RightPart(G));Abs(LineSegmentOf(M, F1))/Abs(LineSegmentOf(M, F2)) = 3;Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 1927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_1927.png", "answer": "x^2-y^2/3=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-a + x)^2 = 3/4);Eccentricity(C) = 2;IsTangent(Asymptote(C),G)", "coords": {} }, { "index": 1928, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1929, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1930, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1930.png", "answer": "4*sqrt(2)/3", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (y = x - 1)", "coords": {} }, { "index": 1931, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1931.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);Slope(LineOf(P, A))=k1;Slope(LineOf(P, B)) = k2;k1:Number;k2:Number;k1*k2 = -4/9", "coords": {} }, { "index": 1932, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1932.png", "answer": "2", "fact_expr": "C: Parabola;F: Point;P:Point;A:Point;Expression(C) = (y^2 = -4*x);Focus(C) = F;Coordinate(A) = (-1, 1);PointOnCurve(P,C)", "coords": { "A": [ -1.0, 1.0 ] } }, { "index": 1933, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1934, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1934.png", "answer": "[2*sqrt(2),+oo]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Inclination(OneOf(Asymptote(G))) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1935, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1935.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Circle;Expression(H) = ((x - sqrt(3))^2 + (y - 1)^2 = 1);IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 1936, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1937, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1937.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;C: Ellipse;P: Point;F: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(l)=(x-y+2=0);LeftFocus(C) = F;PointOnCurve(P, C);Distance(P, l) = d;d:Number", "coords": {} }, { "index": 1938, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1938.png", "answer": "8", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 1939, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1939.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);Slope(LineSegmentOf(P, UpperVertex(C)))*Slope(LineSegmentOf(P, LowerVertex(C))) = -4/9", "coords": {} }, { "index": 1940, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1940.png", "answer": "{16,34}", "fact_expr": "G: Ellipse;t: Number;Expression(G) = (x^2/25 + y^2/t = 1);Focus(G) = {F1,F2};F1: Point;F2: Point;Distance(F1,F2) = 6", "coords": {} }, { "index": 1941, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1941.png", "answer": "12", "fact_expr": "G: Parabola;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A, G);PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B));Focus(G) = F", "coords": {} }, { "index": 1942, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1942.png", "answer": "9/4", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Quadrant(A) = 1;VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 1943, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1944, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.4148966481295064, "b": 2.904512927119681, "orientation": "horizontal" }, "optimization": { "final_loss": 6.037912459509782e-07, "converged": true, "iterations": 470 }, "output_path": "results/test/hyperbola/problem_1944.png", "answer": "7", "fact_expr": "C: Hyperbola;b: Number;b>0;Expression(C)=(x^2-y^2/b^2 = 1);Expression(Asymptote(C))=(y=pm*7*x)", "coords": {} }, { "index": 1945, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1945.png", "answer": "-2", "fact_expr": "T: Ellipse;Expression(T) = (x^2/a^2 + y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(T) = sqrt(2)/2;F: Point;Coordinate(F) = (1, 0);RightFocus(T) = F;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), T) = True;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = M;D: Point;E: Point;M: Point;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, M)) = 1", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 1946, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1947, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1947.png", "answer": "-27/16", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 3*x);PointOnCurve(Focus(G),H);Intersection(H,G) = {A, B}", "coords": {} }, { "index": 1948, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/test/hyperbola/problem_1948.png", "answer": "4*sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (sqrt(3), 2);PointOnCurve(H, G);Eccentricity(G)=3", "coords": { "H": [ 1.7320508075688772, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1949, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1950, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1950.png", "answer": "sqrt(17)/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F2: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l,RightPart(C))={A, B};DotProduct(VectorOf(A,F1),VectorOf(A,F2))=0;Abs(LineSegmentOf(A, F2))/Abs(LineSegmentOf(B, F2)) = 1/2", "coords": {} }, { "index": 1951, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1951.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2;IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 1952, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1953, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1954, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1954.png", "answer": "pm*1", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;k: Number;Expression(C) = (y^2 = 4*x);Slope(l)=k;PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B))=8", "coords": {} }, { "index": 1955, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1956, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1956.png", "answer": "pm*sqrt(2)/4", "fact_expr": "l: Line;C: Parabola;E: Point;F: Point;A: Point;B: Point;k: Number;Expression(C) = (y = x^2/4);Focus(C) = F;Intersection(Directrix(C),axis)=E;PointOnCurve(F, l);Intersection(l, C) = {A, B};3*VectorOf(E, F) = VectorOf(E, A) + 2*VectorOf(E, B);Slope(l) = k", "coords": {} }, { "index": 1957, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1957.png", "answer": "9", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Projection(P, yAxis) = Q;Q: Point;M: Point;Coordinate(M) = (7, 8)", "coords": { "M": [ 7.0, 8.0 ] } }, { "index": 1958, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1958.png", "answer": "(x^2 - y^2/8 = 1)&(x <= -1)", "fact_expr": "C: Circle;Z: Circle;A: Point;P: Point;Expression(C) = (y^2 + (x - 3)^2 = 4);Coordinate(A) = (-3, 0);Center(Z) = P;PointOnCurve(A, Z);IsOutTangent(Z, C)", "coords": { "A": [ -3.0, 0.0 ] } }, { "index": 1959, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1959.png", "answer": "11/2", "fact_expr": "G: Parabola;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 6*x);Focus(G) = F;PointOnCurve(P, G) = True;Coordinate(B) = (4, 3)", "coords": { "B": [ 4.0, 3.0 ] } }, { "index": 1960, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1960.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Circle;Expression(H) = (x^2 + (y + 1)^2 = 1);A: Point;B: Point;TangentPoint(l, G) = A;TangentPoint(l, H) = B;Negation(A=B);F: Point;Focus(G) = F;WhenMin(Abs(LineSegmentOf(A, F)))", "coords": {} }, { "index": 1961, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1962, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1962.png", "answer": "y^2/8+x^2/4=1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Coordinate(F1) = (0, -2);Coordinate(F2) = (0, 2);Focus(G) = {F1, F2};Length(MajorAxis(G)) = 4*sqrt(2)", "coords": { "F1": [ 0.0, -2.0 ], "F2": [ 0.0, 2.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1963, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1963.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;O: Origin;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 1964, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1965, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1965.png", "answer": "5/4", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);F: Point;Focus(G) = F;M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 3", "coords": {} }, { "index": 1966, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1966.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 6)", "coords": {} }, { "index": 1967, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1967.png", "answer": "{y^2=4*x,y^2=16*x}", "fact_expr": "G: Parabola;p: Number;F:Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(F, G);Distance(F,Focus(G)) = 5;Distance(F,SymmetryAxis(G)) = 4", "coords": {} }, { "index": 1968, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1968.png", "answer": "y=pm*2*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (3, 0);OneOf(Focus(G)) = H;Coordinate(OneOf(Vertex(G)))=(1,0)", "coords": { "H": [ 3.0, 0.0 ] } }, { "index": 1969, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1969.png", "answer": "-1", "fact_expr": "l: Line;G: Parabola;F: Point;A: Point;B: Point;P:Point;m:Number;n:Number;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = P;PointOnCurve(P, l);Intersection(l, G) = {A, B};Slope(LineSegmentOf(F, A)) = m;Slope(LineSegmentOf(F,B))=n", "coords": {} }, { "index": 1970, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1971, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1971.png", "answer": "y = pm*x/2", "fact_expr": "C: Hyperbola;a: Number;Expression(C) = (-y^2/5 + x^2/a^2 = 1);a>0;FocalLength(C) = 10", "coords": {} }, { "index": 1972, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1972.png", "answer": "y^2=2*x\n9*sqrt(2)/4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;F: Point;Coordinate(F) = (1/2, 0);Focus(C) = F;P: Point;PointOnCurve(P, C);G: Line;Expression(G) = (x + y + 5 = 0);Q: Point;PointOnCurve(Q, G)", "coords": { "F": [ 0.5, 0.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 1973, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1973.png", "answer": "(1/4,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)", "coords": {} }, { "index": 1974, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1975, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 9.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1975.png", "answer": "4", "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/9 + y^2/7 = 1);Coordinate(A) = (-sqrt(2), 1);LeftFocus(C) = F;PointOnCurve(P, C)", "coords": { "A": [ -1.4142135623730951, 1.0 ] } }, { "index": 1976, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1976.png", "answer": "(0,1)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)", "coords": {} }, { "index": 1977, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1977.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(I) = (sqrt(2), sqrt(6));PointOnCurve(I, OneOf(Asymptote(G)))", "coords": { "I": [ 1.4142135623730951, 2.449489742783178 ] } }, { "index": 1978, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1978.png", "answer": "16/5", "fact_expr": "G: Hyperbola;P: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;Focus(G) = {F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1),LineSegmentOf(P, F2))", "coords": {} }, { "index": 1979, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1979.png", "answer": "15", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/8 = 1);Expression(H) = (x^2/16 + y^2/7 = 1);Focus(G) = {F1, F2};Focus(H) = {F1, F2};OneOf(Intersection(G, H)) = P", "coords": {} }, { "index": 1980, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1980.png", "answer": "sqrt(26)/4", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;Q: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(G)) = {P, Q};IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(P, F1));Abs(LineSegmentOf(P, Q)) = (8/15)*Abs(LineSegmentOf(P, F1))", "coords": {} }, { "index": 1981, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1981.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: LineSegment;IsChordOf(L,G) = True;PointOnCurve(F,L) = True;Intersection(L,G) = {A,B};A: Point;B: Point;L1: Line;L2: Line;PointOnCurve(A,L1) = True;IsPerpendicular(L1,yAxis) = True;PointOnCurve(B,L2) = True;IsPerpendicular(L2,yAxis) = True;FootPoint(L1,yAxis) = C;FootPoint(L2,yAxis) = D;C: Point;D: Point", "coords": {} }, { "index": 1982, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_1982.png", "answer": "x = -4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);Distance(M, Focus(G)) = 5", "coords": {} }, { "index": 1983, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1983.png", "answer": "x^2/9 + y^2/5 = 1", "fact_expr": "O: Circle;G: Circle;A: Point;P: Point;Coordinate(A) = (2, 0);Expression(O) = (x^2 + y^2 = 9);IsDiameter(LineSegmentOf(A, P), G);IsInTangent(G, O)", "coords": {} }, { "index": 1984, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1984.png", "answer": "10", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2}", "coords": {} }, { "index": 1985, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1985.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (0, -4)", "coords": { "Q": [ 0.0, -4.0 ] } }, { "index": 1986, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1986.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 0);Distance(P,OneOf(Asymptote(C)))=1/2", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 1987, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1987.png", "answer": "2*sqrt(10)", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 1988, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_1988.png", "answer": "2;4*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);E: Point;a: Number;Coordinate(E) = (a, 0);l: Line;PointOnCurve(E, l);P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Intersection(l, C) = {P, Q};y1*y2 = -4;M: Point;N: Point;MidPoint(H) = Q;Endpoint(H) = {M, N};PointOnCurve(N, xAxis);PointOnCurve(M, C);H:LineSegment;Negation(P=Q)", "coords": {} }, { "index": 1989, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1989.png", "answer": "x^2-y^2/8=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);RightFocus(C)=F;Expression(Asymptote(C))=(y=pm*(2*sqrt(2)*x));Distance(A, F) = 2*sqrt(2)", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 1990, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1991, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1991.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};l: Line;PointOnCurve(F1, l);M: Point;N: Point;Intersection(l, G) = {M, N}", "coords": {} }, { "index": 1992, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/circle/problem_1992.png", "answer": "y^2=-8*x", "fact_expr": "P: Circle;G: Line;C:Circle;P1:Point;Expression(C)=((x+2)^2+y^2=1);Expression(G) = (x = 1);IsOutTangent(P,C);IsTangent(P,G);Center(P)=P1", "coords": {} }, { "index": 1993, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_1993.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(C))) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1994, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1995, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1996, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1997, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1997.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a^2 = 1);a: Number;a>3;F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;PointOnCurve(F1, LineSegmentOf(A, B)) = True", "coords": {} }, { "index": 1998, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1998.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/7 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 2", "coords": {} }, { "index": 1999, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_1999.png", "answer": "[sqrt(3)/2, 1)", "fact_expr": "C: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);Area(TriangleOf(M, F1, F2)) = sqrt(3)*b^2;Eccentricity(C) = e", "coords": {} }, { "index": 2000, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2000.png", "answer": "pm*1", "fact_expr": "l: Line;G: Ellipse;M: Point;N: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(l)=(y=k*x+1);Intersection(l,G)={M,N};Abs(LineSegmentOf(M, N)) = 4*sqrt(2)/3;k:Number", "coords": {} }, { "index": 2001, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2002, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/test/hyperbola/problem_2002.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 2003, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2003.png", "answer": "2*sqrt(10)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0", "coords": {} }, { "index": 2004, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2004.png", "answer": "(1,3]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Min(Abs(LineSegmentOf(P, F1))^2/Abs(LineSegmentOf(P, F2))) = 8*a;Eccentricity(G) = e", "coords": {} }, { "index": 2005, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2005.png", "answer": "x^2/45+y^2/20=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;L1: Line;L2: Line;Directrix(G) = {L1, L2};Distance(P, L1) = 6;Distance(P, L2) = 12", "coords": {} }, { "index": 2006, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2006.png", "answer": "[1/2,1)", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (y^2/b + x^2/a = 1);RightFocus(C) = F;PointOnCurve(A, C);Max(Distance(A, F))= d;PointOnCurve(P, RightDirectrix(C));LineSegmentOf(P, F) = d;d:Number", "coords": {} }, { "index": 2007, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2007.png", "answer": "[pi/4, pi/3] + [2*pi/3, 3*pi/4]", "fact_expr": "G: Parabola;H: Circle;I: LineSegment;alpha: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (x^2 + y^2 = 3/4);PointOnCurve(Focus(G), I);Inclination(I) = alpha;Negation(Length(I) > 8);IsChordOf(I, G);IsIntersect(OverlappingLine(I), H)", "coords": {} }, { "index": 2008, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2008.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(C) = A;A1: Circle;Center(A1) = A;c: Number;HalfFocalLength(C) = c;Radius(A1) = c;M: Point;N: Point;Intersection(A1, OneOf(Asymptote(C))) = {M, N};Abs(LineSegmentOf(M, N)) = sqrt(3)*c", "coords": {} }, { "index": 2009, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2009.png", "answer": "36", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l1: Line;l2: Line;PointOnCurve(F, l1);PointOnCurve(F, l2);IsPerpendicular(l1, l2);A: Point;B: Point;Intersection(l1, C) = {A, B};D: Point;E: Point;Intersection(l2, C) = {D, E}", "coords": {} }, { "index": 2010, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2010.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P,G) = True;IsPerpendicular(LineSegmentOf(P,F1) , LineSegmentOf(P,F2)) = True", "coords": {} }, { "index": 2011, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2011.png", "answer": "4*sqrt(10)/3", "fact_expr": "l: Line;G: Ellipse;M: Point;A: Point;B: Point;F: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Coordinate(M) = (0, 1);PointOnCurve(M, l);Intersection(l, G) = {A, B};RightFocus(G) = F;WhenMax(Perimeter(TriangleOf(A,B,F)))", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 2012, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_2012.png", "answer": "{0, 1, 4/5}", "fact_expr": "l: Line;G: Curve;a: Real;Expression(G) = (y^2 = a*x);Expression(l) = (y = x*(a - 1) - 1);NumIntersection(l, G) = 1", "coords": {} }, { "index": 2013, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2014, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_2014.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;Directrix(C) = l;M: Point;Coordinate(M) = (1, 0);G: Line;PointOnCurve(M, G);Slope(G) = sqrt(3);A: Point;Intersection(G, l) = A;B: Point;OneOf(Intersection(G, C)) = B;LineSegmentOf(A, M) = LineSegmentOf(M, B)", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 2015, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/test/hyperbola/problem_2015.png", "answer": "10", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = sqrt(2)*x);F: Point;Coordinate(F) = (-sqrt(3), 0);LeftFocus(C) = F;PointOnCurve(F , l) = True;Slope(l) = 2;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), C) = True", "coords": { "F": [ -1.7320508075688772, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2016, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2016.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;m: Number;P: Point;Q: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = -3*m + x);Intersection(G, C) = {P, Q};XCoordinate(MidPoint(LineSegmentOf(P, Q))) = 2*m", "coords": {} }, { "index": 2017, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2017.png", "answer": "4*sqrt(3)/3", "fact_expr": "l: Line;G: Parabola;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};VectorOf(B, A) = 4*VectorOf(B, F)", "coords": {} }, { "index": 2018, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_2018.png", "answer": "x=2", "fact_expr": "C: Parabola;m: Number;E: Circle;Expression(C) = (y^2 = m*x);Expression(E) = (4*x + x^2 + y^2 - 1 = 0);Focus(C) = Center(E)", "coords": {} }, { "index": 2019, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2019.png", "answer": "(sqrt(3)+1)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, LeftPart(C)) = True;Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));AngleOf(P, F1, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 2020, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2020.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Range(Eccentricity(G))=(0, sqrt(2)/2];A: Point;RightFocus(G) = A;B: Point;UpperVertex(G) = B;Distance(LeftFocus(G), LineOf(A, B)) = 2;A1: Point;LeftFocus(G) = A1;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 2021, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2021.png", "answer": "3*x-4*y-7=0", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (1, -1);PointOnCurve(M,l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, -1.0 ] } }, { "index": 2022, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4142164151010692, "b": 1.4142164151010692, "orientation": "horizontal" }, "optimization": { "final_loss": 2.2160979279094996e-06, "converged": true, "iterations": 95 }, "output_path": "results/test/hyperbola/problem_2022.png", "answer": "x^2-y^2/3=1", "fact_expr": "G: Hyperbola;SymmetryAxis(G) = axis;O: Origin;Center(G) = O;F1: Point;F2: Point;Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);Focus(G) = {F1, F2};P: Point;Coordinate(P) = (-2, 3);PointOnCurve(P, G)", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ], "P": [ -2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 2023, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2023.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*(x*(sqrt(3)/2)))", "coords": {} }, { "index": 2024, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2024.png", "answer": "sqrt(3)+sqrt(6)", "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;A: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, RightPart(C));Intersection(LineOf(A,F1), yAxis) = B;Abs(LineSegmentOf(F1, B)) = 3*Abs(LineSegmentOf(A, B));IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2))", "coords": {} }, { "index": 2025, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2025.png", "answer": "(x-2)^2+y^2=4", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2 - y^2/3 = 1);RightFocus(G) = Center(H);Eccentricity(G)=Radius(H)", "coords": {} }, { "index": 2026, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2026.png", "answer": "x^2/9-y^2/7=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 4/3;Focus(G)=Focus(H)", "coords": {} }, { "index": 2027, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2028, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2028.png", "answer": "x+4*y-5=0", "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(I) = (1, 1);MidPoint(H)=I;IsChordOf(H,G)", "coords": { "I": [ 1.0, 1.0 ] } }, { "index": 2029, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_2029.png", "answer": "3/2", "fact_expr": "l: Line;Expression(l) = (-m + x - y = 0);m: Number;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;PointOnCurve(Focus(C), l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 2030, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2030.png", "answer": "(3/4, +oo)", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y = 2*x^2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);PerpendicularBisector(LineSegmentOf(A, B)) = l;Slope(l) = 1/2", "coords": {} }, { "index": 2031, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2031.png", "answer": "10", "fact_expr": "C: Parabola;Expression(C) = (x^2 = -2*p*y);p: Number;p>0;F: Point;Focus(C) = F;Q: Curve;Expression(Q) = (x^2/4 + y^2/8 = 1);OneOf(Focus(Q)) = F;PointOnCurve(F, G) = True;G: Line;Intersection(G, C) = {A, B};Negation(A = B);A: Point;B: Point;Intersection(TangentOnPoint(A, C), TangentOnPoint(B, C)) = M;M: Point;XCoordinate(MidPoint(M)) = 2", "coords": {} }, { "index": 2032, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2032.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Length(LineSegmentOf(A, B)) = 10", "coords": {} }, { "index": 2033, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2033.png", "answer": "sqrt(35)", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 16);Expression(H) = (y = x/2 + 1)", "coords": {} }, { "index": 2034, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2034.png", "answer": "4*sqrt(5)", "fact_expr": "G: Line;H: Curve;A: Point;B: Point;Expression(G) = (y = 2*x + 3);Expression(H) = (y = x^2);Intersection(G, H) = {A, B}", "coords": {} }, { "index": 2035, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2036, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2037, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2037.png", "answer": "3", "fact_expr": "P: Point;Coordinate(P) = (2, a);a: Number;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C) = True", "coords": {} }, { "index": 2038, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2039, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2040, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2040.png", "answer": "(0, 2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (4, 0);RightFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);NumIntersection(H, RightPart(G)) = 1", "coords": { "F": [ 4.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2041, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2041.png", "answer": "8*sqrt(5)/5", "fact_expr": "G: Hyperbola;H: Parabola;C: Circle;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y = x^2/4);Center(C)=Focus(H);Radius(C)=Distance(Focus(H),Directrix(H))", "coords": {} }, { "index": 2042, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2043, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2043.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);Max(AngleOf(F1, P, F2)) = (2*pi)/3;a:Number;b:Number;a>b;b>0", "coords": {} }, { "index": 2044, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2045, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2045.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2046, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2047, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2047.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, Asymptote(C)) = True;O: Origin;Abs(LineSegmentOf(O, P)) = 1;Area(TriangleOf(P, F1, F2)) = sqrt(2)*a", "coords": {} }, { "index": 2048, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2048.png", "answer": "(0, 1/8)", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 2049, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2049.png", "answer": "pm*5", "fact_expr": "G: Hyperbola;P: Point;I: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 - y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);VectorOf(F1, I)*Abs(VectorOf(P, F2)) + VectorOf(P, I)*Abs(VectorOf(F1, F2)) + VectorOf(F2, I)*Abs(VectorOf(P, F1)) = 0", "coords": {} }, { "index": 2050, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2051, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2051.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;B: Point;A: Point;O: Origin;Expression(G) = (x^2 - y^2/2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0", "coords": {} }, { "index": 2052, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2052.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;B: Point;C: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);Negation(A=F);OneOf(Focus(G)) = A;PointOnCurve(B, G);PointOnCurve(C, G);F:Point;OneOf(Focus(G))=F;PointOnCurve(F,LineSegmentOf(B,C))", "coords": {} }, { "index": 2053, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2053.png", "answer": "5", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);FocalLength(G) = 6", "coords": {} }, { "index": 2054, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2054.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,F2),xAxis);Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 2055, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_2055.png", "answer": "(a-p)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);a: Number;Abs(LineSegmentOf(A, B)) = a;a>2*p;M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 2056, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2057, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2057.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;M: Point;N: Point;l1: Line;l2: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);L1:Line;L2:Line;PointOnCurve(P,L1);PointOnCurve(P,L2);Asymptote(G)={l1,l2};IsParallel(L1,l1);IsParallel(L2,l2);Intersection(L1,l2)=M;Intersection(L2,l1)=N;Abs(LineSegmentOf(P, M))*Abs(LineSegmentOf(P, N)) = b^2", "coords": {} }, { "index": 2058, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/test/parabola/problem_2058.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G:Line;M: Point;A: Point;B: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(M) = (1, 0);Directrix(C)=l;PointOnCurve(M,G);Slope(G)=sqrt(3);Intersection(G,l)=A;OneOf(Intersection(G,C))=B;VectorOf(A, M) = VectorOf(M, B)", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 2059, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2059.png", "answer": "6*sqrt(2)/7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);H: Point;Coordinate(H) = (1, 0);l: Line;PointOnCurve(H, l) = True;Inclination(l) = ApplyUnit(45, degree);Intersection(l, G) = {A, B};A: Point;O: Origin;B: Point", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 2060, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3228756555322954, "b": 2.2912878474779204, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2060.png", "answer": "x^2/4-y^2/3=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/16 + y^2/9 = 1);Focus(G) = Focus(H);Eccentricity(G)=2*Eccentricity(H)", "coords": {} }, { "index": 2061, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2062, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2063, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 3.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2063.png", "answer": "{4/7,4}", "fact_expr": "G: Ellipse;C:Ellipse;a: Number;Expression(G) = (a*x^2 + y^2 = 4);Expression(C)=(x^2/12 + y^2/9 = 1);FocalLength(G)=FocalLength(C)", "coords": {} }, { "index": 2064, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2065, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/hyperbola/problem_2065.png", "answer": "(sqrt(2),+oo)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;M:Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;L:Line;OneOf(Asymptote(G))=L;Expression(L)=(y=(b/a)*x);L1:Line;IsPerpendicular(L,L1);FootPoint(L,L1)=M;PointOnCurve(F, L1);Intersection(L1,LeftPart(G)) = A;Intersection(L1,RightPart(G)) = B", "coords": {} }, { "index": 2066, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 22.963727951049805, "direction": "right" }, "optimization": { "final_loss": 107.33567810058594, "converged": false, "iterations": 500 }, "output_path": "results/test/parabola/problem_2066.png", "answer": "y^2 = -16*x", "fact_expr": "G: Parabola;H: Point;O: Origin;Coordinate(H) = (-4, -8);PointOnCurve(Focus(G), xAxis);PointOnCurve(H, G);Vertex(G) = O", "coords": { "H": [ -4.0, -8.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 2067, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/ellipse/problem_2067.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 2068, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/test/parabola/problem_2068.png", "answer": "32", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);A: Point;B: Point;Intersection(G, C) = {A, B}", "coords": {} } ] }