{ "stats": { "total": 1035, "success": 788, "failed": 247 }, "type_stats": { "ellipse": 220, "hyperbola": 293, "parabola": 262, "circle": 13 }, "reason_counts": { "hyperbola_point_off_curve": 3, "hyperbola_focus_mismatch": 5, "parabola_point_off_curve": 3, "circle_point_off_curve": 1, "ellipse_ecc_mismatch": 4, "ellipse_point_off_curve": 2, "ellipse_focus_mismatch": 1 }, "results": [ { "index": 0, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0000.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Circle;F1: Point;Z: Line;T: Point;P: Point;M: Point;Expression(G) = (x^2/16 - y^2/25 = 1);LeftFocus(G) = F1;Expression(H) = (x^2 + y^2 = 16);TangentOfPoint(F1, H) = Z;TangentPoint(Z, H) = T;Intersection(OverlappingLine(LineSegmentOf(F1, T)), RightPart(G)) = P;MidPoint(LineSegmentOf(F1, P)) = M;O: Origin", "coords": {} }, { "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/dev/ellipse/problem_0001.png", "answer": "[-2, 1]", "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": 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/dev/parabola/problem_0002.png", "answer": "(1, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 3, "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/dev/hyperbola/problem_0003.png", "answer": "x^2-y^2/3=1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightVertex(G) = F;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)", "coords": {} }, { "index": 4, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5, "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/dev/hyperbola/problem_0005.png", "answer": "x^2/3-y^2/9=1", "fact_expr": "Gamma: Hyperbola;P: Point;Coordinate(P) = (sqrt(6), 3);Expression(Asymptote(Gamma))=(y=pm*sqrt(3)*x);PointOnCurve(P,Gamma)", "coords": { "P": [ 2.449489742783178, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 6, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 0.0 ], "radius": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0006.png", "answer": "x^2/9+y^2/8=1", "fact_expr": "G: Circle;Expression(G) = (y^2 + (x + 1)^2 = 36);Center(G) = M;M: Point;A: Point;PointOnCurve(A, G);N: Point;Coordinate(N) = (1, 0);Intersection(PerpendicularBisector(LineSegmentOf(A,N)),LineSegmentOf(M,A)) = P;P: Point", "coords": {} }, { "index": 7, "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/dev/ellipse/problem_0007.png", "answer": "1", "fact_expr": "G: Ellipse;O: Circle;R: Number;F: Point;P: Point;Expression(G) = (x^2/4 + y^2/2 = 1);R>0;Expression(O) = (x^2 + y^2 = R^2);OneOf(Focus(G)) = F;OneOf(Endpoint(MinorAxis(G)))=P;IsTangent(LineOf(P,F),O)", "coords": {} }, { "index": 8, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 9, "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/dev/parabola/problem_0009.png", "answer": "8", "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(Directrix(C), xAxis) = M;Tan(AngleOf(A, M, B)) = 2*sqrt(2)", "coords": {} }, { "index": 10, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0010.png", "answer": "sqrt(3)+1", "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));DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 11, "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/dev/ellipse/problem_0011.png", "answer": "2*x+2*y-3=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);M: Point;Coordinate(M) = (1, 1/2);PointOnCurve(M, H) = True;H: Line;Intersection(H, G) = {P, Q};Q: Point;P: Point;MidPoint(LineSegmentOf(P, Q)) = M", "coords": { "M": [ 1.0, 0.5 ] } }, { "index": 12, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0012.png", "answer": "(0,1/2)", "fact_expr": "l: Line;P: Parabola;p: Number;B: Point;C: Point;F: Point;O: Origin;M: Point;N: Point;p>0;Expression(P) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(P) = F;PointOnCurve(F, l);k:Number;Slope(l) = k;Intersection(l, P) = {B, C};L:Line;PointOnCurve(O,L);MidPoint(LineSegmentOf(B,C))=M;PointOnCurve(M,L);Intersection(L,P)=N", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 13, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0013.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;p: Number;l: Line;H:Line;M: Point;N: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G)=l;PointOnCurve(F, H);Inclination(H)=ApplyUnit(60,degree);Intersection(H,G)=M;Quadrant(M) = 1;IsPerpendicular(LineSegmentOf(M,N),l);FootPoint(LineSegmentOf(M,N),l)=N;Area(TriangleOf(M, N, F)) = 4*sqrt(3)", "coords": {} }, { "index": 14, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 15, "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/dev/ellipse/problem_0015.png", "answer": "144/5", "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/25 + y^2/16 = 1);Coordinate(A) = (0, 36/5);RightFocus(C) = F;PointOnCurve(P, C);WhenMax(Perimeter(TriangleOf(A,P,F)))", "coords": { "A": [ 0.0, 7.2 ] } }, { "index": 16, "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/dev/parabola/problem_0016.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "l: Line;C: Parabola;A: Point;Expression(C) = (y^2 = 6*x);Coordinate(A) = (0, 2);PointOnCurve(A, l);IsTangent(l, C)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 17, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 18, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0018.png", "answer": "5/4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = (3/4)*x)", "coords": {} }, { "index": 19, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/parabola/problem_0020.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Point;L:LineSegment;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(H) = (3, 1);IsChordOf(L,G);MidPoint(L)=H;Slope(L)=2", "coords": { "H": [ 3.0, 1.0 ] } }, { "index": 21, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 2.8284271247461903 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0021.png", "answer": "x^2/2+y^2=1", "fact_expr": "C: Circle;B: Point;A: Point;P: Point;Expression(C) = (y^2 + (x - 1)^2 = 8);Coordinate(A) = (-1, 0);PointOnCurve(B, C);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), LineSegmentOf(B, C)) = P", "coords": {} }, { "index": 22, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0022.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/3 - y^2/3 = 1)", "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/dev/parabola/problem_0023.png", "answer": "1/2", "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);PointOnCurve(F,LineOf(A,B));-Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 3/2", "coords": {} }, { "index": 24, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0024.png", "answer": "x = -1", "fact_expr": "H: Circle;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;IsTangent(H, Directrix(G)) = True", "coords": {} }, { "index": 25, "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/dev/parabola/problem_0025.png", "answer": "14", "fact_expr": "G: Parabola;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G),LineSegmentOf(A,B));IsChordOf(LineSegmentOf(A,B),G);x1 + x2 = 10", "coords": {} }, { "index": 26, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 27, "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/dev/parabola/problem_0027.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;E: Point;Intersection(Directrix(C), xAxis) = E;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;Abs(LineSegmentOf(M, E)) = sqrt(11)", "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/dev/hyperbola/problem_0028.png", "answer": "3*sqrt(5)/5", "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;Expression(G) = (-c + x = 0);c: Number;P: Point;OneOf(Intersection(G, C)) = P;Q: Point;Intersection(G, OneOf(Asymptote(C))) = Q;O: Origin;VectorOf(O, P) = (1/3)*VectorOf(O, F2) + (2/3)*VectorOf(O, Q)", "coords": {} }, { "index": 29, "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/dev/ellipse/problem_0029.png", "answer": "(0,sqrt(3)/2]", "fact_expr": "C: Ellipse;b: Number;G: Line;k: Number;b>0;Expression(C) = (x^2/4 + y^2/b^2 = 1);Expression(G) = (y = k*x - 1);PointOnCurve(Focus(C),xAxis);IsIntersect(G,C)", "coords": {} }, { "index": 30, "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/dev/parabola/problem_0030.png", "answer": "4", "fact_expr": "M: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G) = True;F: Point;Focus(G) = F;A: Point;C: Circle;Expression(C) = ((x - 4)^2 + (y - 1)^2 = 1);PointOnCurve(A, C) = True", "coords": {} }, { "index": 31, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.8425120115280151, "b": 1.6167607307434082, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3765338735538535e-05, "converged": true, "iterations": 54 }, "output_path": "results/dev/hyperbola/problem_0031.png", "answer": "3", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/7 + x^2/a^2 = 1);Eccentricity(G) = 4/3", "coords": {} }, { "index": 32, "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/dev/ellipse/problem_0032.png", "answer": "11", "fact_expr": "G: Ellipse;M: Point;N: Point;P: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (1, 4);Coordinate(N) = (2, 0);PointOnCurve(P, G)", "coords": { "M": [ 1.0, 4.0 ], "N": [ 2.0, 0.0 ] } }, { "index": 33, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0033.png", "answer": "sqrt(107)/4", "fact_expr": "E: Ellipse;Expression(E) = (x^2/8 + y^2/2 = 1);P: Point;PointOnCurve(P, E);A: Point;B: Point;Coordinate(A) = (-2, 1);Coordinate(B) = (2, -1);l: Line;Expression(l) = (x=4);M: Point;N: Point;Intersection(LineOf(A, P), l) = M;Intersection(LineOf(B, P), l) = N;Area(TriangleOf(A, B, P)) = Area(TriangleOf(M, N, P));O: Origin", "coords": { "A": [ -2.0, 1.0 ], "B": [ 2.0, -1.0 ] } }, { "index": 34, "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/dev/ellipse/problem_0034.png", "answer": "sqrt(7)", "fact_expr": "G: Ellipse;b: Number;F: Point;P:Point;m:Number;M:Number;0 < b;b < 4;Expression(G) = (x^2/16 + y^2/b^2 = 1);LeftFocus(G) = F;PointOnCurve(P,G);Max(Length(LineSegmentOf(F,P)))=M;Min(Length(LineSegmentOf(F,P)))=m;M = 7*m", "coords": {} }, { "index": 35, "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/dev/parabola/problem_0035.png", "answer": "4*sqrt(6)", "fact_expr": "E: Parabola;C: Circle;a: Number;Expression(E) = (x^2 = 4*y);Expression(C) = (a*y + 8*x + x^2 + y^2 - 5 = 0);PointOnCurve(Focus(E), C)", "coords": {} }, { "index": 36, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0036.png", "answer": "(sqrt(5)+1)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (0, b);RightFocus(G) = F;IsPerpendicular(OverlappingLine(LineSegmentOf(B,F)),OneOf(Asymptote(G)))", "coords": {} }, { "index": 37, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0037.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - 9*y^2 = 9);Expression(H) = (2*x - 6*y - 21 = 0)", "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/dev/hyperbola/problem_0038.png", "answer": "sqrt(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;m: Number;m>0;Coordinate(OneOf(Focus(G))) = (m, 0);P: Point;Coordinate(P) = (m, 2*m);PointOnCurve(P, G)", "coords": {} }, { "index": 39, "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/dev/parabola/problem_0039.png", "answer": "6", "fact_expr": "F: Point;C: Parabola;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(F, l);l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;l1: Line;l2: Line;TangentOnPoint(A, C) = l1;TangentOnPoint(B, C) = l2;Intersection(l1, l2)= P;P: Point;Negation(A=B)", "coords": {} }, { "index": 40, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 41, "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/dev/ellipse/problem_0041.png", "answer": "3", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;Distance(M, F1) = 4;MidPoint(LineSegmentOf(M, F1)) = N", "coords": {} }, { "index": 42, "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/dev/ellipse/problem_0042.png", "answer": "12", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(F1, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A,B),G);Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 8;RightFocus(G) = F2", "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/dev/hyperbola/problem_0043.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;A: Point;B: Point;P: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A=B);Negation(A=P);Negation(B=P);O: Origin;PointOnCurve(O, LineSegmentOf(A, B));k1: Number;k2: Number;Slope(LineOf(P, A)) = k1;Slope(LineOf(P, B)) = k2;k1*k2=3", "coords": {} }, { "index": 44, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0044.png", "answer": "2*sqrt(3)/3", "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))) = (2*pi)/3;Eccentricity(C)=e;e:Number", "coords": {} }, { "index": 45, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 46, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.046616818012331, "b": 3.9644919697671575, "orientation": "horizontal" }, "optimization": { "final_loss": 4.0501961192983754e-07, "converged": true, "iterations": 93 }, "output_path": "results/dev/hyperbola/problem_0046.png", "answer": "3, 15", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(F1) = (-5, 0);Coordinate(F2) = (5, 0);PointOnCurve(P, G);Distance(P, F1) = 9", "coords": { "F1": [ -5.0, 0.0 ], "F2": [ 5.0, 0.0 ] } }, { "index": 47, "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/dev/ellipse/problem_0047.png", "answer": "5", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2 + y^2/m = 1);FocalLength(G) = 4", "coords": {} }, { "index": 48, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0048.png", "answer": "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);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F2,L);IsPerpendicular(L,OneOf(Asymptote(C)));FootPoint(L,OneOf(Asymptote(C)))=P;Abs(LineSegmentOf(P, F1)) = sqrt(5)*a ;O:Origin", "coords": {} }, { "index": 49, "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/dev/hyperbola/problem_0049.png", "answer": "2+2*sqrt(3)", "fact_expr": "C: Hyperbola;P: Point;F2: Point;F1: Point;x1:Number;y1:Number;Expression(C) = (x^2/3 - y^2 = 1);Coordinate(P) = (x1, y1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F2)) = 2", "coords": {} }, { "index": 50, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0050.png", "answer": "25", "fact_expr": "T: Point;Coordinate(T) = (-1, 1);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;L1: Line;L2: Line;TangentOfPoint(T, C) = {L1, L2};TangentPoint(L1, C) = A;TangentPoint(L2, C) = B;A: Point;B: Point;Focus(C) = F;F: Point;PointOnCurve(F, LineOf(A, B)) = True", "coords": { "T": [ -1.0, 1.0 ] } }, { "index": 51, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 52, "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/dev/ellipse/problem_0052.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;M: 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;VectorOf(A, M) = VectorOf(M, F2);DotProduct(VectorOf(M, F1), VectorOf(A, F2)) = 0", "coords": {} }, { "index": 53, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 54, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 55, "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/dev/ellipse/problem_0055.png", "answer": "2", "fact_expr": "G: Ellipse;O: Origin;M: Point;P: Point;F: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 10;LeftFocus(G) = F;VectorOf(O, M) = (VectorOf(O, F) + VectorOf(O, P))/2", "coords": {} }, { "index": 56, "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/dev/ellipse/problem_0056.png", "answer": "x+2*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/4 + y^2/2 = 1);IsChordOf(H,G);Coordinate(MidPoint(H))=(1,1)", "coords": {} }, { "index": 57, "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/dev/hyperbola/problem_0057.png", "answer": "y=pm*(1/2)*x\nx^2/4-y^2=1", "fact_expr": "G: Hyperbola;a: Number;b: Number;F1: Point;P: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};Eccentricity(G)=sqrt(5)/2;PointOnCurve(P, G);AngleOf(F1, P,F2) = ApplyUnit(90, degree);Area(TriangleOf(F1,P,F2))=1", "coords": {} }, { "index": 58, "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/dev/parabola/problem_0058.png", "answer": "x^2+y^2-2*x=0", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Circle;Focus(G) = Center(H);O: Origin;PointOnCurve(O, H)", "coords": {} }, { "index": 59, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 60, "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/dev/hyperbola/problem_0060.png", "answer": "11/21", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 61, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0061.png", "answer": "sqrt(10)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F1: 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(F2, l);Intersection(l, RightPart(C)) = {A, B};DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0;3*VectorOf(A, F2) = VectorOf(F2, B)", "coords": {} }, { "index": 62, "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/dev/ellipse/problem_0062.png", "answer": "sqrt(5)/7", "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);F1: Point;F2: Point;Focus(G) = {F1,F2};AngleOf(P, F1, F2) = alpha;AngleOf(P, F2, F1) = beta;alpha: Number;beta: Number;Cos(alpha) = sqrt(5)/5;Sin(alpha + beta) = 3/5", "coords": {} }, { "index": 63, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 64, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 65, "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/dev/ellipse/problem_0065.png", "answer": "sqrt(7)/3", "fact_expr": "F1: Point;F2: Point;Focus(E) = {F1, F2};E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;PointOnCurve(A, E) = True;AngleOf(F1, A, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(A,F1)) = 2*Abs(LineSegmentOf(A,F2))", "coords": {} }, { "index": 66, "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/dev/ellipse/problem_0066.png", "answer": "pi/4", "fact_expr": "C: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(C) = (x^2/4 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, H);Intersection(H, C) = {A, B}", "coords": {} }, { "index": 67, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 68, "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/dev/parabola/problem_0068.png", "answer": "32", "fact_expr": "T: Parabola;Expression(T) = (y^2 = 8*x);F: Point;Focus(T) = F;l1: Line;l2: Line;PointOnCurve(F, l1);PointOnCurve(F, l2);IsPerpendicular(l1, l2);A: Point;B: Point;C: Point;D: Point;Intersection(l1, T) = {A, B};Intersection(l2, T) = {C, D};P: Point;Q: Point;MidPoint(LineSegmentOf(A, B)) = P;MidPoint(LineSegmentOf(C, D)) = Q", "coords": {} }, { "index": 69, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 70, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0070.png", "answer": "4", "fact_expr": "G: Ellipse;l: Line;F1: Point;A: Point;B: Point;F2:Point;Expression(G) = (4*x^2 + y^2 = 1);Focus(G)={F1,F2};PointOnCurve(F1,l);Intersection(l,G)={A,B}", "coords": {} }, { "index": 71, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.830951894845301, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 34.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0071.png", "answer": "y^2/9 - x^2/16 = 1", "fact_expr": "S: Hyperbola;G: Ellipse;Expression(G) = (x^2/9 + y^2/34 = 1);Focus(S)=Focus(G);Expression(OneOf(Asymptote(S)))=(y=(3/4)*x)", "coords": {} }, { "index": 72, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0072.png", "answer": "pm*x - 2*y = 0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 1)", "coords": {} }, { "index": 73, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/parabola/problem_0074.png", "answer": "sqrt(15)", "fact_expr": "C: Parabola;l: Line;Expression(C) = (y^2 = 4*x);Expression(l) = (2*x - y - 1 = 0)", "coords": {} }, { "index": 75, "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/dev/parabola/problem_0075.png", "answer": "3*sqrt(2) - 1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;Expression(l) = (x - y + 5 = 0);P: Point;PointOnCurve(P, G) = True;d1: Number;Distance(P, yAxis) = d1;d2: Number;Distance(P,l) = d2", "coords": {} }, { "index": 76, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 77, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 78, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 79, "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/dev/parabola/problem_0079.png", "answer": "4/3", "fact_expr": "l: Line;C: Parabola;A: Point;D: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l, Directrix(C)) = D;MidPoint(LineSegmentOf(A, D)) = F", "coords": {} }, { "index": 80, "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/dev/ellipse/problem_0080.png", "answer": "5/7", "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);P: Point;Q: Point;Intersection(H, G) = {P, Q};LineSegmentOf(P, F2) = LineSegmentOf(F1, F2);3*LineSegmentOf(P, F1) = 4*LineSegmentOf(Q, F1)", "coords": {} }, { "index": 81, "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/dev/hyperbola/problem_0081.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)", "coords": {} }, { "index": 82, "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/dev/hyperbola/problem_0082.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 83, "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/dev/ellipse/problem_0083.png", "answer": "[-7,8]", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/45 + y^2/20 = 1);PointOnCurve(P, G) = True;Quadrant(P) = 3;J1: Point;J2: Point;Focus(G) = {J1, J2};IsPerpendicular(LineSegmentOf(P, J1), LineSegmentOf(P, J2)) = True;H: Line;Expression(H) = (-2*m + 4*x - 3*y + 1 = 0);Negation(Distance(P, H) > 3);m: Real", "coords": {} }, { "index": 84, "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/dev/ellipse/problem_0084.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x - 2*y + 2 = 0);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)", "coords": {} }, { "index": 85, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0085.png", "answer": "x^2/5-y^2/4=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;C: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (-6*x + x^2 + y^2 + 5 = 0);IsTangent(Asymptote(G),C);Center(C)=RightFocus(G)", "coords": {} }, { "index": 86, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0086.png", "answer": "y=pm*2*x", "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);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,C);Abs(VectorOf(P,F1)+VectorOf(P,F2))=Abs(VectorOf(F1,F2));Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 87, "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/dev/parabola/problem_0087.png", "answer": "(5*sqrt(5))/2", "fact_expr": "H: Point;Coordinate(H) = (1, 0);l: Line;PointOnCurve(H, l);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Coordinate(C) = (-1, -1);IsPerpendicular(LineSegmentOf(C, A), LineSegmentOf(C, B))", "coords": { "H": [ 1.0, 0.0 ], "C": [ -1.0, -1.0 ] } }, { "index": 88, "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/dev/parabola/problem_0088.png", "answer": "y=pm*2*x", "fact_expr": "C: Parabola;N: Point;O: Origin;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = M;PointOnCurve(N, C);lambda:Number;Abs(VectorOf(M,N))=lambda*Abs(VectorOf(N,F));WhenMax(lambda)", "coords": {} }, { "index": 89, "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/dev/parabola/problem_0089.png", "answer": "{0,pm*1/2}", "fact_expr": "l: Line;G: Parabola;k: Real;Expression(G) = (y^2 = x);Expression(l) = (y = k*(x + 1));NumIntersection(l,G)=1", "coords": {} }, { "index": 90, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0090.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Line;O: Origin;A: Point;B: Point;C: Point;Expression(G) = (x^2/12 - y^2/3 = 1);Expression(H) = (y = x*(sqrt(3)/3) - 2);Intersection(H, RightPart(G)) = {A, B};PointOnCurve(C, RightPart(G));VectorOf(O, A) + VectorOf(O, B) = t*VectorOf(O, C);t:Number", "coords": {} }, { "index": 91, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 92, "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/dev/hyperbola/problem_0092.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 93, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 94, "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/dev/parabola/problem_0094.png", "answer": "2", "fact_expr": "C: Parabola;A: Point;B: Point;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (2, 2);Focus(C) = F;PointOnCurve(A, C);PointOnCurve(B, C);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 95, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 96, "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/dev/parabola/problem_0096.png", "answer": "32/3", "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": 97, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 98, "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/dev/ellipse/problem_0098.png", "answer": "x^2+y^2/(2/3)=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = sqrt(3)/3;l: Line;PointOnCurve(F2, l);A: Point;B: Point;Intersection(l, C) = {A, B};Perimeter(TriangleOf(A, F1, B)) = 4", "coords": {} }, { "index": 99, "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/dev/parabola/problem_0099.png", "answer": "3/2 - sqrt(2)", "fact_expr": "G: Parabola;M: Point;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (0, 1);Focus(G) = F;Intersection(LineSegmentOf(F, M), G) = A", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 100, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 101, "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/dev/hyperbola/problem_0101.png", "answer": "4/5", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;I: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(I,P,F2))=Area(TriangleOf(I,P,F1))-lambda*Area(TriangleOf(I,F1,F2));lambda:Number", "coords": {} }, { "index": 102, "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/dev/parabola/problem_0102.png", "answer": "48", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);H: Line;PointOnCurve(Focus(G), H);Inclination(H) = ApplyUnit(60, degree);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 103, "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/dev/circle/problem_0103.png", "answer": "(x - 3/2)^2 + y^2 = 1/4", "fact_expr": "B: Point;Coordinate(B) = (3, 0);A: Point;G: Circle;Expression(G) = (x^2 + y^2 = 1);PointOnCurve(A, G) = True;M: Point;MidPoint(LineSegmentOf(A,B)) = M", "coords": {} }, { "index": 104, "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/dev/ellipse/problem_0104.png", "answer": "x^2/4+y^2=1", "fact_expr": "l: Line;E: Ellipse;b: Number;a: Number;A: Point;F: Point;B: Point;O:Origin;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F;Eccentricity(E)=sqrt(3)/2;PointOnCurve(O,l);Inclination(l)=pi/3;Intersection(l,E)={A,B};Perimeter(TriangleOf(A, F, B))=4+(8*sqrt(13)/13)", "coords": {} }, { "index": 105, "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/dev/parabola/problem_0105.png", "answer": "5*sqrt(2)/2-1", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;L1:Line;L2:Line;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y + 4 = 0);PointOnCurve(P, G);PointOnCurve(P,L1);PointOnCurve(P,L2);IsPerpendicular(L1,yAxis);IsPerpendicular(L2,H);FootPoint(L1,yAxis)=A;FootPoint(L2,H)=B", "coords": {} }, { "index": 106, "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/dev/hyperbola/problem_0106.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Line;A: Point;F2: Point;B: 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)) = {A, B}", "coords": {} }, { "index": 107, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 108, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 109, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 2.0 ], "radius": 3.872983346207417 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0109.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/a = 1);a: Number;H: Circle;Expression(H) = (-4*y - 2*x + x^2 + y^2 - 10 = 0);PointOnCurve(Center(H), Asymptote(G))", "coords": {} }, { "index": 110, "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/dev/ellipse/problem_0110.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;B: Point;C: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);PointOnCurve(F2,LineSegmentOf(B,C));PointOnCurve(B, G);PointOnCurve(C, G);F1:Point;F2:Point;Focus(G)={F1,F2};A=F1", "coords": {} }, { "index": 111, "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/dev/parabola/problem_0111.png", "answer": "4", "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;F: Point;Expression(C) = (y = x^2/4);Focus(C) = F;PointOnCurve(F, l);Inclination(l)=pi/4;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 112, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 113, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 114, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0114.png", "answer": "sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = ApplyUnit(120, degree);IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = M;M: Point;L: Line;PointOnCurve(M, L);IsPerpendicular(L, l);Directrix(G) = l;l: Line;FootPoint(L, l) = M1;M1: Point", "coords": {} }, { "index": 115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0115.png", "answer": "5", "fact_expr": "C: Parabola;p: Number;F: Point;P: Point;A: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (4, 2);Coordinate(F) = (1, 0);PointOnCurve(P, C);Focus(C) = F", "coords": { "A": [ 4.0, 2.0 ], "F": [ 1.0, 0.0 ] } }, { "index": 116, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0116.png", "answer": "3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;P0: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (0, 3*b);PointOnCurve(P, l);L1: Line;OneOf(Asymptote(C)) = L1;Slope(L1) > 0;IsParallel(l, L1);PointOnCurve(P0, RightPart(C));Distance(P0, l) > b", "coords": {} }, { "index": 117, "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/dev/parabola/problem_0117.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F:Point;Expression(G) = (y^2 = 4*x);Inclination(l)=ApplyUnit(45,degree);Focus(G)=F;PointOnCurve(F,l);Intersection(l,G)={A,B}", "coords": {} }, { "index": 118, "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/dev/ellipse/problem_0118.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;F1: Point;F2: Point;Focus(G) = {F1, F2};Z1: Line;Z2: Line;Directrix(G) = {Z1, Z2};M: Point;N: Point;Intersection(Z1, xAxis) = M;Intersection(Z2, xAxis) = N;LineSegmentOf(M, N) <= 2*LineSegmentOf(F1, F2)", "coords": {} }, { "index": 119, "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/dev/parabola/problem_0119.png", "answer": "x = -4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);H: Point;Coordinate(H) = (4, 0);PointOnCurve(H, G) = True;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point", "coords": { "H": [ 4.0, 0.0 ] } }, { "index": 120, "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/dev/circle/problem_0120.png", "answer": "{(1/3,+oo),(-oo,-1/3)}", "fact_expr": "G: Hyperbola;k: Real;H: Circle;Expression(G) = (-1/(4*k^2)*y^2 + x^2/((9*k^2)) = 1);Expression(H) = (x^2 + y^2 = 1);NumIntersection(G, H)=0", "coords": {} }, { "index": 121, "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/dev/ellipse/problem_0121.png", "answer": "5/7", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;M: Point;F2: Point;F1: Point;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {M, N};Abs(LineSegmentOf(M, F2)) = Abs(LineSegmentOf(F1, F2));7*Abs(LineSegmentOf(M, F1)) = 4*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 122, "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/dev/hyperbola/problem_0122.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 32", "coords": {} }, { "index": 123, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0123.png", "answer": "5/3", "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, F1)) = 4*Abs(LineSegmentOf(P, F2));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 124, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0124.png", "answer": "pm*sqrt(3)*a", "fact_expr": "G: Parabola;a: Number;a>0;Expression(G) = (y^2 = 2*(a*x));H:Point;PointOnCurve(H,G);Distance(H,Focus(G))=2*a", "coords": {} }, { "index": 125, "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/dev/ellipse/problem_0125.png", "answer": "(0,pm*1)", "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) = 5*VectorOf(F2, B)", "coords": {} }, { "index": 126, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0126.png", "answer": "-4", "fact_expr": "l: Line;p: Number;A: Point;B: Point;p>0;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};G: Parabola", "coords": {} }, { "index": 127, "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/dev/parabola/problem_0127.png", "answer": "10", "fact_expr": "G: Parabola;M: Point;P: 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": 128, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 129, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0129.png", "answer": "5", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;O: Origin;M: Point;PointOnCurve(M, LeftPart(G));Abs(LineSegmentOf(O, M)) = sqrt(a^2 + b^2);4*Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 130, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 131, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 132, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 133, "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/dev/hyperbola/problem_0133.png", "answer": "y=pm*2*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 134, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 135, "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/dev/ellipse/problem_0135.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;B: Point;A: Point;J: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(J) = (1, 1/2);PointOnCurve(Focus(G), xAxis);L1:Line;L2:Line;TangentOfPoint(J,H)={L1,L2};TangentPoint(L1,H)=A;TangentPoint(L2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))", "coords": { "J": [ 1.0, 0.5 ] } }, { "index": 136, "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/dev/parabola/problem_0136.png", "answer": "9", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F,G) = True;Intersection(G, C) = {A, B};A: Point;B: Point", "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/dev/ellipse/problem_0137.png", "answer": "x + 2*y - 2 = 0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);P: Point;Coordinate(P) = (1, 1/2);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 0.5 ] } }, { "index": 138, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0138.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;L: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, L);IsPerpendicular(L, Asymptote(C));FootPoint(L, Asymptote(C)) = A;Intersection(L, yAxis) = B;VectorOf(F, A) = 2*VectorOf(A, B)", "coords": {} }, { "index": 139, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0139.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;p: Real;G: Circle;l:Line;A: Point;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Directrix(C)=l;Intersection(l,xAxis)=A;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);IsDiameter(LineSegmentOf(A,F),G);Length(InterceptChord(LineOf(A,P),G))=2", "coords": {} }, { "index": 140, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0140.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/7 - y^2/9 = -1)", "coords": {} }, { "index": 141, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 142, "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/dev/parabola/problem_0142.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, C) = True;Quadrant(P) = 1;PointOnCurve(P, G) = True;IsPerpendicular(G, l) = True;G: Line;FootPoint(G, l) = M;M: Point;Slope(LineOf(P, F)) = sqrt(3)", "coords": {} }, { "index": 143, "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/dev/parabola/problem_0143.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 5", "coords": {} }, { "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/dev/parabola/problem_0144.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(Directrix(G)) = (x = -1)", "coords": {} }, { "index": 145, "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/dev/parabola/problem_0145.png", "answer": "(3, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x)", "coords": {} }, { "index": 146, "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/dev/ellipse/problem_0146.png", "answer": "(sqrt(5)-1)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: 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": 147, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0147.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H) = True;Intersection(H, G) = {P, Q};P: Point;Q: Point;IsPerpendicular(LineSegmentOf(P, H1), Directrix(G)) ;IsPerpendicular(LineSegmentOf(P, H2), Directrix(G)) ;FootPoint(LineSegmentOf(P, H1), Directrix(G)) = H1;FootPoint(LineSegmentOf(P, H2), Directrix(G)) = H2;PointOnCurve(P, LineSegmentOf(P, H1));PointOnCurve(Q, LineSegmentOf(P, H2));H1: Point;H2: Point;MidPoint(LineSegmentOf(H1, H2)) = M;M: Point;Abs(LineSegmentOf(P, F)) = a;a: Number;Abs(LineSegmentOf(Q, F)) = b;b: Number", "coords": {} }, { "index": 148, "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/dev/hyperbola/problem_0148.png", "answer": "4*sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/6 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = 2", "coords": {} }, { "index": 149, "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/dev/ellipse/problem_0149.png", "answer": "2*x+4*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/2 + y^2 = 1);Coordinate(MidPoint(H)) = (1/2, 1/2);IsChordOf(H,G) = True", "coords": {} }, { "index": 150, "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/dev/parabola/problem_0150.png", "answer": "2", "fact_expr": "G: Line;Expression(G) = (2*x + y - 3 = 0);Gamma: Parabola;Expression(Gamma) = (y^2 = 8*x);A: Point;B: Point;Intersection(G, Gamma) = {A, B};H: Circle;PointOnCurve(A, H);PointOnCurve(B, H);C: Point;D: Point;Intersection(H, Gamma) = {C, D};k: Number;Slope(LineOf(C, D)) = k", "coords": {} }, { "index": 151, "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/dev/parabola/problem_0151.png", "answer": "2*sqrt(3)/3", "fact_expr": "O: Origin;F: Point;Vertex(G) = O;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);M: Point;PointOnCurve(M, G)", "coords": {} }, { "index": 152, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0152.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (2*x + x^2 + y^2 - 3 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 153, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 154, "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/dev/ellipse/problem_0154.png", "answer": "9", "fact_expr": "G: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "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/dev/hyperbola/problem_0155.png", "answer": "(1, 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: Circle;Expression(H) = (y^2 + (x - sqrt(3))^2 = 2);IsIntersect(Asymptote(G), H)", "coords": {} }, { "index": 156, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0156.png", "answer": "y^2=4*x", "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) = True;Inclination(l) = pi/4;Intersection(l,G) = {A,B};A: Point;B: Point;IsDiameter(LineSegmentOf(A,B),H) = True;H: Circle;PointOnCurve(I,H) = True;I: Point;Coordinate(I) = (-p/2, 2)", "coords": {} }, { "index": 157, "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/dev/hyperbola/problem_0157.png", "answer": "(4/9)*x^2 - x^2/4 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (2*sqrt(3), -3);PointOnCurve(A, C);Asymptote(C) = Asymptote(G)", "coords": { "A": [ 3.4641016151377544, -3.0 ] } }, { "index": 158, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 159, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0159.png", "answer": "sqrt(5)/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/2)", "coords": {} }, { "index": 160, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 161, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 162, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 163, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0163.png", "answer": "[sqrt(3),+oo)", "fact_expr": "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;F2: Point;F1: Point;PointOnCurve(P,C) = True;P: Point;DotProduct(VectorOf(P,F1),VectorOf(P,F2)) = -2*a^2", "coords": {} }, { "index": 164, "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/dev/hyperbola/problem_0164.png", "answer": "x^2/9+y^2/5=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);Vertex(G) = Focus(H);Focus(G) = Vertex(H);H: Ellipse", "coords": {} }, { "index": 165, "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/dev/ellipse/problem_0165.png", "answer": "5/2", "fact_expr": "C: Ellipse;M: Point;F: Point;O: Point;N: Point;Expression(C) = (x^2/25 + y^2/9 = 1);LeftFocus(C) = F;PointOnCurve(M, C);MidPoint(LineSegmentOf(M, F)) = N;Center(C) = O;LineSegmentOf(O, N) = 4", "coords": {} }, { "index": 166, "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/dev/ellipse/problem_0166.png", "answer": "{4, sqrt(34)}", "fact_expr": "G: Ellipse;m: Real;m>0;Expression(G) = (x^2/25 + y^2/m^2 = 1);FocalLength(G)=6", "coords": {} }, { "index": 167, "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/dev/ellipse/problem_0167.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/12 + y^2/3 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 168, "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/dev/ellipse/problem_0168.png", "answer": "x^2/72+y^2/36=1", "fact_expr": "M: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;D: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-c, 0);Coordinate(D) = (-4, 2);Eccentricity(M) = sqrt(2)/2;LeftFocus(M)=F;PointOnCurve(F, G);Intersection(G, M) = {A, B};IsChordOf(LineSegmentOf(A, B),M);MidPoint(LineSegmentOf(A, B)) = D;c:Number", "coords": { "D": [ -4.0, 2.0 ] } }, { "index": 169, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0169.png", "answer": "8/5", "fact_expr": "A: Point;Coordinate(A) = (4, 0);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;0 < p;p < 4;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, H), l) = True;FootPoint(LineSegmentOf(P, H), l) = H;H: Point;Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree)", "coords": { "A": [ 4.0, 0.0 ] } }, { "index": 170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0170.png", "answer": "-7*sqrt(15)/17", "fact_expr": "C1: Hyperbola;C2: Circle;A: Point;B: Point;F2: Point;F1: Point;m: Number;m > 0;Expression(C1) = (x^2/9 - y^2/m = 1);LeftFocus(C1) = F1;RightFocus(C1) = F2;Expression(Asymptote(C1)) = (sqrt(7)*x + pm*3*y = 0);PointOnCurve(A, C2);Expression(C2) = (x^2 + y^2 = 16);VectorOf(F1, A) = VectorOf(A, B);PointOnCurve(B, RightPart(C1))", "coords": {} }, { "index": 171, "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/dev/hyperbola/problem_0171.png", "answer": "sqrt(2) / 2", "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "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/dev/parabola/problem_0172.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);IsChordOf(LineSegmentOf(A,B),G);Focus(G) = F;PointOnCurve(F,LineSegmentOf(A,B));Abs(LineSegmentOf(A, B)) = 8;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 173, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 174, "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/dev/ellipse/problem_0174.png", "answer": "sqrt(10)/5", "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) = (-1, 0);Coordinate(F2) = (1, 0);LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;Expression(G) = (x + y - 2 = 0);A: Point;PointOnCurve(A, G);PointOnCurve(A, C)", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ] } }, { "index": 175, "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/dev/parabola/problem_0175.png", "answer": "4", "fact_expr": "G: Parabola;M: Point;x0:Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (x0, 2*sqrt(3));PointOnCurve(M,G)", "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/dev/hyperbola/problem_0176.png", "answer": "sqrt(5)", "fact_expr": "P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;PointOnCurve(P, G) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 177, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 178, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0178.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;A1: Point;A2: Point;F: Point;P: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;LeftVertex(C) = A1;RightVertex(C) = A2;PointOnCurve(F, l);L1: Line;L2: Line;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1=L2);IsParallel(l, L1);Intersection(l, L2) = P;IsDiameter(LineSegmentOf(A1, A2), G);PointOnCurve(P, G)", "coords": {} }, { "index": 179, "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/dev/parabola/problem_0179.png", "answer": "y^2/4 - x^2/5 = 1", "fact_expr": "C: Hyperbola;O: Origin;Center(C) = O;G: Parabola;Expression(G) = (y = x^2/12);OneOf(Focus(C)) = Focus(G);Expression(OneOf(Asymptote(C))) = (y = x*(2*sqrt(5)/5))", "coords": {} }, { "index": 180, "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/dev/hyperbola/problem_0180.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightVertex(G) = F;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)", "coords": {} }, { "index": 181, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0181.png", "answer": "pm*sqrt(3)", "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 182, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 183, "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/dev/ellipse/problem_0183.png", "answer": "(3-sqrt(5))/2", "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;A: Point;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;UpperVertex(C) = B;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,LineSegmentOf(A,B));IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))", "coords": {} }, { "index": 184, "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/dev/ellipse/problem_0184.png", "answer": "2", "fact_expr": "G: Point;Coordinate(G) = (2, 3);PointOnCurve(G, C);C: Hyperbola;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = 4", "coords": { "G": [ 2.0, 3.0 ] } }, { "index": 185, "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/dev/parabola/problem_0185.png", "answer": "3", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 186, "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/dev/parabola/problem_0186.png", "answer": "1", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);F: Point;Focus(C) = F;P: Point;y0: Number;Coordinate(P) = (1, y0);PointOnCurve(P, C);A: Point;l: Line;TangentOnPoint(P,C)=l;Intersection(l, yAxis) = A", "coords": {} }, { "index": 187, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 188, "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/dev/parabola/problem_0188.png", "answer": "(1/8,-1)", "fact_expr": "P: Point;Coordinate(P) = (4, -1);G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;M: Point;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, P)) + Abs(LineSegmentOf(M, F)))", "coords": { "P": [ 4.0, -1.0 ] } }, { "index": 189, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0189.png", "answer": "sqrt(5)", "fact_expr": "F: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;OneOf(Focus(C)) = F;P: Point;PointOnCurve(P, C);MidPoint(LineSegmentOf(P, F)) = OneOf(Endpoint(ImageinaryAxis(C)))", "coords": {} }, { "index": 190, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 191, "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/dev/parabola/problem_0191.png", "answer": "11/2", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;F: Point;B: Point;m>0;Expression(C) = (y^2 = 4*x);Coordinate(M) = (m, 0);Focus(C)=F;PointOnCurve(F, G);Intersection(G, C) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);Abs(VectorOf(M, F)) = Abs(VectorOf(M, A));m:Number", "coords": {} }, { "index": 192, "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/dev/ellipse/problem_0192.png", "answer": "15", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;P: Point;F1: Point;Coordinate(A) = (-1, 3);LeftFocus(G) = F1;PointOnCurve(P, G)", "coords": { "A": [ -1.0, 3.0 ] } }, { "index": 193, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 194, "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/dev/parabola/problem_0194.png", "answer": "6", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (4, 1);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ 4.0, 1.0 ] } }, { "index": 195, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 196, "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/dev/ellipse/problem_0196.png", "answer": "6", "fact_expr": "C: Ellipse;A: Point;B: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftVertex(C) = A;UpperVertex(C) = B;RightFocus(C) = F", "coords": {} }, { "index": 197, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0197.png", "answer": "x^2/5 - y^2 = 1", "fact_expr": "C: Hyperbola;b: Number;A: Point;O: Origin;M: Point;N: Point;b>0;A0: Circle;Expression(C) = (x^2/5 - y^2/b^2 = 1);RightVertex(C) = A;Center(A0) = A;Radius(A0) = b;Intersection(A0, OneOf(Asymptote(C))) = {M, N};VectorOf(O, M) = (3/2)*VectorOf(O, N)", "coords": {} }, { "index": 198, "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/dev/hyperbola/problem_0198.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;P: Point;A: Point;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, 2);Coordinate(F) = (-sqrt(5), 0);LeftFocus(G) = F;PointOnCurve(P, RightPart(G));Min(Perimeter(TriangleOf(P, A, F))) = 8", "coords": { "A": [ 0.0, 2.0 ], "F": [ -2.23606797749979, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 199, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 200, "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/dev/parabola/problem_0200.png", "answer": "6", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;C: Point;D: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(C) = (0, 3);Coordinate(D) = (4, 0);PointOnCurve(C,H);Intersection(H,G)={A,B};PointOnCurve(D,PerpendicularBisector(LineSegmentOf(A,B)));Focus(G)=F", "coords": { "C": [ 0.0, 3.0 ], "D": [ 4.0, 0.0 ] } }, { "index": 201, "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/dev/ellipse/problem_0201.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 202, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0203.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/b^2 = 1);b: Number;e: Number;Eccentricity(G) = e;e = sqrt(2)", "coords": {} }, { "index": 204, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 205, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 206, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0206.png", "answer": "4", "fact_expr": "l: Line;Slope(l) > 0;Sin(Inclination(l)) = sqrt(2)/2;Intercept(l, xAxis) = -2;Intersection(l, C) = {A, B};A: Point;B: Point;C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;Abs(LineSegmentOf(A, B)) = 16", "coords": {} }, { "index": 207, "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/dev/hyperbola/problem_0207.png", "answer": "x^2-y^2/3=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;m: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/(m^2 + 4) + y^2/m^2 = 1);Focus(H) = Focus(G);Eccentricity(G) = 2", "coords": {} }, { "index": 208, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 209, "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/dev/ellipse/problem_0209.png", "answer": "{(1,4)+(4,oo)}", "fact_expr": "G: Ellipse;n: Number;H: Line;k: Real;Expression(G) = (x^2/4 + y^2/n = 1);Expression(H) = (y = k*x + 1);NumIntersection(H, G) = 2", "coords": {} }, { "index": 210, "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/dev/ellipse/problem_0210.png", "answer": "(1,sqrt(10)/2]", "fact_expr": "Gamma: Ellipse;H: Circle;I: Line;N: Point;M: Point;O:Origin;Expression(I) = (x + y = 1);Expression(Gamma) = (y^2/b^2 + x^2/a^2 = 1);Intersection(I, Gamma) = {M, N};IsDiameter(LineSegmentOf(M,N),H);PointOnCurve(O,H);Negation(Eccentricity(Gamma)>sqrt(3)/2);a:Number;b:Number;a>b;b>0", "coords": {} }, { "index": 211, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0211.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Circle;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(M) = (y^2 + (x - 2)^2 = 3);IsTangent(Asymptote(C),M);Eccentricity(C) = e", "coords": {} }, { "index": 212, "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/dev/hyperbola/problem_0212.png", "answer": "(pm*sqrt(7), 0)", "fact_expr": "C: Hyperbola;Expression(C) = (3*x^2 - 4*y^2 = 12)", "coords": {} }, { "index": 213, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 214, "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/dev/hyperbola/problem_0214.png", "answer": "3", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/9 + y^2/16 = 1);F: Point;RightFocus(C) = F", "coords": {} }, { "index": 215, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 216, "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/dev/ellipse/problem_0216.png", "answer": "4", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);M: Point;PointOnCurve(M, C);Distance(M, RightFocus(C)) = 2", "coords": {} }, { "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/dev/parabola/problem_0217.png", "answer": "1/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;C: Point;D: Point;PointOnCurve(Focus(G), LineSegmentOf(A, B));PointOnCurve(Focus(G), LineSegmentOf(C, D));IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D));IsChordOf(LineSegmentOf(A, B), G);IsChordOf(LineSegmentOf(C, D), G)", "coords": {} }, { "index": 218, "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/dev/ellipse/problem_0218.png", "answer": "4/5", "fact_expr": "C: Ellipse;A: Point;P: Point;B: Point;Q: Point;M: Point;N: Point;Expression(C) = (x^2/4 + y^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(M, xAxis);l:Line;PointOnCurve(M,l);Intersection(l, C) = {P, Q};l1:Line;PointOnCurve(M, l1);IsPerpendicular(l1,LineSegmentOf(A, P));Intersection(l1, LineSegmentOf(B, Q)) = N;IsPerpendicular(l,xAxis)", "coords": {} }, { "index": 219, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 220, "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/dev/hyperbola/problem_0220.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2 - y^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 221, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0221.png", "answer": "y^2=8*x", "fact_expr": "H: Line;Slope(H) = 2;G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;F: Point;Focus(G) = F;PointOnCurve(F, H) = True;Intersection(H, yAxis) = A;O: Origin;A: Point;Area(TriangleOf(O, A, F)) = 4", "coords": {} }, { "index": 222, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 223, "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/dev/parabola/problem_0223.png", "answer": "-3", "fact_expr": "l: Line;G: Parabola;O: Origin;A: Point;B: Point;Expression(G) = (y = x^2);Expression(l) = (y=2*x+m);TangentPoint(l,G)=A;Intersection(l, yAxis) = B;m: Number", "coords": {} }, { "index": 224, "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/dev/hyperbola/problem_0224.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1)", "coords": {} }, { "index": 225, "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/dev/ellipse/problem_0225.png", "answer": "[-sqrt(5),sqrt(5)]", "fact_expr": "G: Ellipse;x: Real;Expression(G) = (x^2/5 + y^2/4 = 1)", "coords": {} }, { "index": 226, "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/dev/hyperbola/problem_0226.png", "answer": "1/4", "fact_expr": "G: Hyperbola;B: Point;A: Point;Expression(G) = (x^2/4 - y^2 = 1);PointOnCurve(A,Asymptote(G));PointOnCurve(B,Asymptote(G));Coordinate(MidPoint(LineSegmentOf(A,B)))=(2,2)", "coords": {} }, { "index": 227, "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/dev/hyperbola/problem_0227.png", "answer": "y^2=-12*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G)=Focus(H)", "coords": {} }, { "index": 228, "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/dev/hyperbola/problem_0228.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 229, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 230, "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/dev/ellipse/problem_0230.png", "answer": "{(0,-1),(-12/11,-7/11)}", "fact_expr": "Gamma: Ellipse;H: Line;A: Point;C: Point;B: Point;Expression(H) = (x - 3*y + 1 = 0);Expression(Gamma)=(x^2/2+y^2=1);Intersection(H,Gamma)={A,B};PointOnCurve(C,Gamma);AngleOf(A, C, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 231, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0231.png", "answer": "{x^2=-8*y,y^2=16*x}", "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (x - 2*y - 4 = 0);Vertex(G) = O;PointOnCurve(Focus(G), H);SymmetryAxis(G)=axis", "coords": {} }, { "index": 232, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 25.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0232.png", "answer": "x^2/18-y^2/2=1", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/5 = 1);G: Hyperbola;Focus(H) = Focus(G);I: Point;Coordinate(I) = (-3*sqrt(2), 0);PointOnCurve(I, G)", "coords": { "I": [ -4.242640687119286, 0.0 ] } }, { "index": 233, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 234, "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/dev/ellipse/problem_0234.png", "answer": "sqrt(2)", "fact_expr": "L: Line;Slope(L) = sqrt(2)/2;G: Ellipse;Expression(G) = (x^2 + y^2/b^2 = 1);b: Number;b>0;PointOnCurve(Focus(G), xAxis);P: Point;Q: Point;Negation(P=Q);Intersection(L, G) = {P, Q};F1: Point;F2: Point;Focus(G) = {F1, F2};Projection(P, xAxis) = F1;Projection(Q, xAxis) = F2", "coords": {} }, { "index": 235, "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/dev/ellipse/problem_0235.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (y^2/9 + x^2/a^2 = 1);a: Number;a>3;F2: Point;F1: Point;Focus(G) = {F1,F2};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;Abs(LineSegmentOf(F1,F2)) = 8;PointOnCurve(F1, LineSegmentOf(A, B))", "coords": {} }, { "index": 236, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 237, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.5030499696731567, "b": 2.00545072555542, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3176339734854992e-06, "converged": true, "iterations": 109 }, "output_path": "results/dev/hyperbola/problem_0237.png", "answer": "y = pm*(4/3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 5/3", "coords": {} }, { "index": 238, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 239, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.449489742783178, "major_axis": "x", "x_coef": 8.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0239.png", "answer": "3/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/8 + y^2/6 = 1);P: Point;Coordinate(P) = (-2, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ -2.0, 1.0 ] } }, { "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/dev/ellipse/problem_0240.png", "answer": "5/7", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2, LineOf(P, F1)) = Length(MinorAxis(C))", "coords": {} }, { "index": 241, "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/dev/ellipse/problem_0241.png", "answer": "20", "fact_expr": "G: Ellipse;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G)=F1;PointOnCurve(F1, H);Intersection(H, G)={A,B};RightFocus(G) = F2", "coords": {} }, { "index": 242, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 243, "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/dev/ellipse/problem_0243.png", "answer": "5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 244, "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/dev/ellipse/problem_0244.png", "answer": "5", "fact_expr": "C: Ellipse;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(C) = (y^2/16 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Perimeter(TriangleOf(P, F1, F2)) = 16", "coords": {} }, { "index": 245, "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/dev/parabola/problem_0245.png", "answer": "sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), xAxis) = M;P: Point;PointOnCurve(P, G);Q: Point;MidPoint(LineSegmentOf(P, F)) = Q", "coords": {} }, { "index": 246, "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/dev/ellipse/problem_0246.png", "answer": "x^2/4+y^2=1", "fact_expr": "G: Parabola;H: Ellipse;b: Number;a: Number;Expression(G) = (y^2 = 4*(sqrt(3)*x));a>0;b>0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G));Length(MajorAxis(H)) = 2*Length(MinorAxis(H))", "coords": {} }, { "index": 247, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 248, "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/dev/ellipse/problem_0248.png", "answer": "x^2/16+y^2/4=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(G) = sqrt(3)/2;H: Line;Slope(H) = 1;M: Point;Coordinate(M) = (b, 0);PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 32/(5*Tan(AngleOf(A, O, B)))", "coords": {} }, { "index": 249, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 250, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0250.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;p > 0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 251, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0251.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;C: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, l);Slope(l) > 0;Intersection(l, G) = {A, B};Quadrant(A) = 1;Intersection(l,Directrix(G)) = C;Abs(LineSegmentOf(B, C)) = 3*Abs(LineSegmentOf(B, F))", "coords": {} }, { "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/dev/parabola/problem_0252.png", "answer": "y = x - 1", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;k: Number;k > 0;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = k;Intersection(l, C) = {A, B};LineSegmentOf(A, B) = 8", "coords": {} }, { "index": 253, "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/dev/parabola/problem_0253.png", "answer": "6", "fact_expr": "C: Parabola;G: Circle;M: Point;A: Point;B: Point;F: Point;a: Number;b: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (a, b);PointOnCurve(M, C);Focus(C) = F;Distance(M, F) = 5;Center(G) = M;PointOnCurve(F, G);Intersection(G, yAxis) = {A, B}", "coords": {} }, { "index": 254, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0254.png", "answer": "4", "fact_expr": "H: Ellipse;Expression(H) = (x^2/5 + (16*y^2)/p^2 = 1);p: Number;G: Parabola;Expression(G) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(H), Directrix(G)) = True", "coords": {} }, { "index": 255, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0255.png", "answer": "[sqrt(10), +oo)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/b^2 = 1);b: Number;b >= 9", "coords": {} }, { "index": 256, "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/dev/parabola/problem_0256.png", "answer": "9*sqrt(3)/4", "fact_expr": "C: Parabola;A: Point;F: Point;O: Origin;N: Point;l: Line;Expression(C) = (y^2 = 6*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(A, C);Distance(A, l) = 6;Intersection(PerpendicularBisector(LineSegmentOf(A, F)), l) = N", "coords": {} }, { "index": 257, "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/dev/ellipse/problem_0257.png", "answer": "-10", "fact_expr": "l: Line;C: Ellipse;F: Point;M: Point;B: Point;A: Point;Expression(C) = (x^2/5 + y^2 = 1);RightFocus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l, yAxis) =M;PointOnCurve(B, LineSegmentOf(F,M))", "coords": {} }, { "index": 258, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0258.png", "answer": "((x-2)^2 + y^2 = 4)&Negation(x = 4)", "fact_expr": "L: Line;O: Circle;M: Point;N: Point;G: Point;Expression(O) = (x^2 + y^2 = 36);Expression(L) = (y = k*(x - 4));Intersection(L, O) = {M, N};MidPoint(LineSegmentOf(M, N)) = G", "coords": {} }, { "index": 259, "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/dev/ellipse/problem_0259.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);00;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = 4", "coords": {} }, { "index": 264, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0264.png", "answer": "40", "fact_expr": "C: Parabola;p: Number;G: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + 2*y - 2 = 0);PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 265, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 266, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 267, "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/dev/ellipse/problem_0267.png", "answer": "x+2*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(P) = (1, 1);IsChordOf(H, G);PointOnCurve(P, H);MidPoint(H) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 268, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 269, "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/dev/parabola/problem_0269.png", "answer": "x^2+y^2=4", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 4*x);Center(H) = Vertex(G);Radius(H) = Distance(Focus(G), Directrix(G))", "coords": {} }, { "index": 270, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0270.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;l: Line;Expression(l) = (x - 2*y - sqrt(5) = 0);IsParallel(OneOf(Asymptote(G)), l) = True;PointOnCurve(OneOf(Focus(G)), l)", "coords": {} }, { "index": 271, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0271.png", "answer": "sqrt(6)/2", "fact_expr": "Gamma: Hyperbola;Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;M: Point;Coordinate(M) = (1, 1);H: Line;Slope(H) = 1/2;PointOnCurve(M, H);A: Point;B: Point;Intersection(H, Gamma) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 272, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0272.png", "answer": "sqrt(10)/2", "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;O: Origin;A: Point;PointOnCurve(A, E);AngleOf(F1, A, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(O, A)) = b", "coords": {} }, { "index": 273, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 274, "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/dev/parabola/problem_0274.png", "answer": "8", "fact_expr": "G: Parabola;F: Point;A: 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;Slope(LineOf(A, F)) = -sqrt(3)", "coords": {} }, { "index": 275, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0275.png", "answer": "sqrt(7)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;k: Number;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = k*x);LeftFocus(C) = F1;RightFocus(C) = F2;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));AngleOf(F1, A, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 276, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 277, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0277.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;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);PointOnCurve(P, G);Negation(A = B);Negation(A = P);Negation(B = P);PointOnCurve(O, LineSegmentOf(A, B));kPA: Number;kPB: Number;Slope(LineOf(P, A)) = kPA;Slope(LineOf(P, B)) = kPB;kPA*kPB = 3", "coords": {} }, { "index": 278, "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/dev/parabola/problem_0278.png", "answer": "(3/2,17/8)", "fact_expr": "M: Point;Coordinate(M) = (3/2, -1);l:Line;C:Parabola;Expression(C)=(x^2=4*y);A: Point;B: Point;PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};IsTangent(LineSegmentOf(A,M),C)", "coords": { "M": [ 1.5, -1.0 ] } }, { "index": 279, "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/dev/parabola/problem_0279.png", "answer": "-3", "fact_expr": "G: Parabola;Expression(G) = (y^2=4*x);H: Point;Coordinate(H) = (1, 0);L: Line;PointOnCurve(H,L) = True;Intersection(L,G) = {P,Q};P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;Q: Point;Coordinate(Q) = (x2,y2);x2: Number;y2: Number", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 280, "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/dev/parabola/problem_0280.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);C: Circle;Expression(C) = ((x + 3)^2 + (y - 3)^2 = 1);P: Point;Q: Point;R: Point;PointOnCurve(P, G);PointOnCurve(Q, C);Projection(P, yAxis) = R", "coords": {} }, { "index": 281, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 282, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0282.png", "answer": "1", "fact_expr": "C: Ellipse;M: Circle;r: Number;P: Point;A: Point;B: Point;Expression(C) = (x^2/8 + y^2/2 = 1);0 < r;r < sqrt(2);Expression(M) = (-r^2 - 4*sqrt(2)*y + 2*(sqrt(2)*x) + x^2 + y^2 + 10 = 0);UpperVertex(C)=P;L1:Line;L2:Line;TangentOfPoint(P,M)={L1,L2};Intersection(L1,C)=A;Intersection(L2,C)=B;Negation(A=P);Negation(B=P)", "coords": {} }, { "index": 283, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 284, "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/dev/hyperbola/problem_0284.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4)", "coords": {} }, { "index": 285, "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/dev/hyperbola/problem_0285.png", "answer": "x - y - 3 = 0", "fact_expr": "G: Hyperbola;H: LineSegment;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (4, 1);MidPoint(H) = P;IsChordOf(H, G)", "coords": { "P": [ 4.0, 1.0 ] } }, { "index": 286, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 287, "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/dev/ellipse/problem_0287.png", "answer": "x^2/9+y^2/8=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;C1: Circle;Expression(C1) = (x^2 + y^2 = 9);NumIntersection(C,C1) = 2;C2: Circle;Expression(C2) = (x^2 + y^2 = 8);NumIntersection(C,C2) = 2", "coords": {} }, { "index": 288, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 289, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0289.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Circle;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(M) = (y^2 + (x - 3)^2 = 8);Intersection(OneOf(Asymptote(C)),M) = {A, B};Abs(LineSegmentOf(A, B)) = 2*sqrt(2)", "coords": {} }, { "index": 290, "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/dev/hyperbola/problem_0290.png", "answer": "y^2=16*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);RightVertex(G)=Focus(H)", "coords": {} }, { "index": 291, "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/dev/hyperbola/problem_0291.png", "answer": "10;y = pm*(4/3)*x;4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 292, "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/dev/hyperbola/problem_0292.png", "answer": "(x^2+y^2=1)&(Negation(y=0))", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G));G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Q: Point;lambda: Number;DotProduct(VectorOf(Q, F2), (VectorOf(P, F1)/Abs(VectorOf(P, F1)) + VectorOf(P, F2)/Abs(VectorOf(P, F2)))) = 0;VectorOf(Q, P) + lambda*(VectorOf(P, F1)/Abs(VectorOf(P, F1)) + VectorOf(P, F2)/Abs(VectorOf(P, F2))) = 0", "coords": {} }, { "index": 293, "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/dev/parabola/problem_0293.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Line;A: Point;Q: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);PointOnCurve(A, H);Intersection(H, G) = {P, Q};MidPoint(LineSegmentOf(A, Q)) = P", "coords": { "A": [ -1.0, 0.0 ], "B": [ 1.0, 0.0 ] } }, { "index": 294, "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/dev/ellipse/problem_0294.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);0 < x0^2/2+y0^2;x0^2/2+y0^2 < 1", "coords": {} }, { "index": 295, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 296, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 297, "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/dev/ellipse/problem_0297.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 298, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0298.png", "answer": "(1,2]", "fact_expr": "G: Hyperbola;a: Number;b: 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));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e", "coords": {} }, { "index": 299, "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/dev/ellipse/problem_0299.png", "answer": "y^2 - x^2 / 3 = 1", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (y^2 + x^2/a = 1);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 300, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.1622776601683795, "b": 3.1622776601683795, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0300.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 10)", "coords": {} }, { "index": 301, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 302, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 303, "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/dev/ellipse/problem_0303.png", "answer": "1/3", "fact_expr": "G: Ellipse;a: Number;b: Number;c: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};PointOnCurve(P, G);Area(TriangleOf(P, F1, F2)) = b^2*(sqrt(2)/2)", "coords": {} }, { "index": 304, "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/dev/hyperbola/problem_0304.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;H: Line;P: Point;Q: Point;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(OneOf(Focus(G)), H);IsPerpendicular(H, xAxis);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=P;Intersection(H,L2)=Q", "coords": {} }, { "index": 305, "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/dev/parabola/problem_0305.png", "answer": "5/2", "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};O: Origin;Abs(LineSegmentOf(A, F)) = 5", "coords": {} }, { "index": 306, "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/dev/hyperbola/problem_0306.png", "answer": "(pm*2,0)", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(C))=2;Eccentricity(C) = 2", "coords": {} }, { "index": 307, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0307.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;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/2;IsChordOf(LineSegmentOf(A, B), G);M: Point;MidPoint(LineSegmentOf(A, B)) = M;N: Point;Projection(M, Directrix(G)) = N", "coords": {} }, { "index": 308, "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/dev/ellipse/problem_0308.png", "answer": "4*sqrt(3)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/2 = 1);l: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(l) = (x - 2*y + 1 = 0);LeftFocus(C) = F1;RightFocus(C) = F2;Intersection(l, C) = {P, Q}", "coords": {} }, { "index": 309, "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/dev/ellipse/problem_0309.png", "answer": "[0, sqrt(2)]", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;O: Origin", "coords": {} }, { "index": 310, "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/dev/ellipse/problem_0310.png", "answer": "1", "fact_expr": "l: Line;G: Ellipse;P: Point;Q: Point;A: Point;B: Point;O: Origin;m: Number;n: Number;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 1);Coordinate(Q) = (m, n);PointOnCurve(P, l);Intersection(l, G) = {A, B};lambda: Number;VectorOf(A, P) = lambda*VectorOf(P, B);VectorOf(A, Q) = -lambda*VectorOf(Q, B)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 311, "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/dev/parabola/problem_0311.png", "answer": "9", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 8", "coords": {} }, { "index": 312, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 313, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 314, "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/dev/ellipse/problem_0314.png", "answer": "2/a", "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": 315, "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/dev/ellipse/problem_0315.png", "answer": "3*x-4*y+7=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;Coordinate(P) = (-1, 1);H: LineSegment;IsChordOf(H, G);MidPoint(H) = P", "coords": { "P": [ -1.0, 1.0 ] } }, { "index": 316, "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/dev/ellipse/problem_0316.png", "answer": "16", "fact_expr": "G: Ellipse;H: Line;M: Point;N: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {M, N}", "coords": {} }, { "index": 317, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 318, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0318.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F2: Point;A: Point;F1: Point;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));IsParallel(LineOf(A,F2),H);Perimeter(TriangleOf(A, F1, F2)) = 9*a", "coords": {} }, { "index": 319, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 320, "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/dev/hyperbola/problem_0320.png", "answer": "8*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1)", "coords": {} }, { "index": 321, "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/dev/hyperbola/problem_0321.png", "answer": "8*sqrt(3)/3", "fact_expr": "G: Hyperbola;H: Line;F: Point;P: Point;Q: Point;Expression(G) = (x^2 - y^2 = 4);RightFocus(G)=F;PointOnCurve(F,H);Inclination(H)=ApplyUnit(105,degree);Intersection(H,G)={P,Q}", "coords": {} }, { "index": 322, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 323, "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/dev/hyperbola/problem_0323.png", "answer": "12/13", "fact_expr": "C: Hyperbola;P: Point;F2: Point;F1: Point;Expression(C) = (x^2/16 - y^2/20 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);LineSegmentOf(P, F2) = 5", "coords": {} }, { "index": 324, "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/dev/parabola/problem_0324.png", "answer": "64", "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 16*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(30, degree);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 325, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 326, "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/dev/ellipse/problem_0326.png", "answer": "1/3", "fact_expr": "H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 1);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Vertex(G)), H);PointOnCurve(OneOf(Focus(G)), H);e: Number;Eccentricity(G) = e", "coords": {} }, { "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/dev/parabola/problem_0327.png", "answer": "x=-4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;G: Line;Expression(G) = (x + 2*y - 4 = 0);PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 328, "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/dev/hyperbola/problem_0328.png", "answer": "x^2/3-y^2/6=1", "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;Focus(C) = {F1, F2};Eccentricity(C) = sqrt(3);P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2*sqrt(3)", "coords": {} }, { "index": 329, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0329.png", "answer": "4*sqrt(7)/7", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;O: Origin;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);RightFocus(G) = Focus(H);Center(G) = O", "coords": {} }, { "index": 330, "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/dev/parabola/problem_0330.png", "answer": "5", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2=4*x);PointOnCurve(P,G) = True;F: Point;Focus(G) = F;A: Point;Coordinate(A) = (4,a);a: Number;Abs(a) < 4", "coords": {} }, { "index": 331, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 332, "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/dev/parabola/problem_0332.png", "answer": "4*sqrt(2)", "fact_expr": "l: Line;C: Parabola;A: Point;O: Origin;B: Point;k1: Number;k2: Number;Expression(C) = (y^2 = 4*x);Intersection(C, l) = {A, B};Slope(LineOf(O, A)) = k1;Slope(LineOf(O, B)) = k2;k1*k2 = -2", "coords": {} }, { "index": 333, "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/dev/parabola/problem_0333.png", "answer": "(0,-2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -8*y)", "coords": {} }, { "index": 334, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 335, "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/dev/parabola/problem_0335.png", "answer": "(4, 4)", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 5;Quadrant(M)=1", "coords": {} }, { "index": 336, "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/dev/parabola/problem_0336.png", "answer": "(x - 1)^2 + (y + pm*2*sqrt(2))^2 = 9", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);A: Point;PointOnCurve(A, C) ;G: Circle;Center(G) = A;PointOnCurve(Vertex(C), G) ;PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 337, "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/dev/hyperbola/problem_0337.png", "answer": "5/3\n4", "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/9 - y^2/16 = 1);Eccentricity(G) = e", "coords": {} }, { "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/dev/hyperbola/problem_0338.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;F: Point;a > 0;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;Slope(H) = 1;Intersection(H, RightPart(G)) = B;Projection(B, xAxis) = F;RightFocus(G) = F;PointOnCurve(A, H)", "coords": {} }, { "index": 339, "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/dev/parabola/problem_0339.png", "answer": "5", "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (m, 4);PointOnCurve(A,G);m:Number", "coords": {} }, { "index": 340, "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/dev/hyperbola/problem_0340.png", "answer": "2*sqrt(3)/3", "fact_expr": "C1: Hyperbola;b: Number;a: Number;C2: Hyperbola;a>0;b>0;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);Expression(C2) = (y^2/b^2 - x^2/a^2 = 1);Eccentricity(C1) = 2", "coords": {} }, { "index": 341, "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/dev/parabola/problem_0341.png", "answer": "8", "fact_expr": "G: Parabola;B: Point;A: Point;P:Point;Expression(G) = (y = x^2/16);PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(A, B)) = 24;MidPoint(LineSegmentOf(A,B))=P;WhenMin(Distance(P,xAxis))", "coords": {} }, { "index": 342, "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/dev/ellipse/problem_0342.png", "answer": "-1/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);l: Line;PointOnCurve(OneOf(Focus(G)), l);Inclination(l) = ApplyUnit(45, degree);A: Point;B: Point;Intersection(l, G) = {A, B};O: Origin", "coords": {} }, { "index": 343, "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/dev/ellipse/problem_0343.png", "answer": "7", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 4)^2 = 4);G: Ellipse;Expression(G) = (x^2 + y^2/4 = 1);P: Point;Q: Point;PointOnCurve(P,C) = True;PointOnCurve(Q,G) = True", "coords": {} }, { "index": 344, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0344.png", "answer": "{y^2=8*x, y^2=-16*x}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;H: Line;Expression(H) = (x = 1);Distance(Directrix(G), H) = 3", "coords": {} }, { "index": 345, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0345.png", "answer": "12", "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (-y^2/3 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = x/2)", "coords": {} }, { "index": 346, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 347, "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/dev/ellipse/problem_0347.png", "answer": "3*sqrt(3)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/12 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 348, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0348.png", "answer": "sqrt(2)", "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;L: Line;PointOnCurve(F,L)=True;IsPerpendicular(Asymptote(G),L)=True;FootPoint(Asymptote(G),L) = P;P: Point;PointOnCurve(MidPoint(LineSegmentOf(P,F)),G) = True", "coords": {} }, { "index": 349, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 350, "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/dev/parabola/problem_0350.png", "answer": "sqrt(10)/5", "fact_expr": "l: Line;C: Parabola;E:Circle;E1: Point;O: Origin;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(E1) = (1, 1);Slope(l)=1/3;Intersection(l,C)={A,B};Center(E)=E1;InscribedCircle(TriangleOf(O,A,B))=E", "coords": { "E1": [ 1.0, 1.0 ] } }, { "index": 351, "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/dev/ellipse/problem_0351.png", "answer": "x+2*y-2=0", "fact_expr": "P: Point;Coordinate(P) = (1, 1);H: LineSegment;G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);IsChordOf(H, G) = True;MidPoint(H) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 352, "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/dev/hyperbola/problem_0352.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 353, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0353.png", "answer": "y=pm*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;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);DotProduct((VectorOf(O,P)+VectorOf(O,F2)),VectorOf(F2,P))=0;Abs(VectorOf(P, F1))*Abs(VectorOf(P, F2)) = 2*a^2", "coords": {} }, { "index": 354, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 355, "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/dev/parabola/problem_0355.png", "answer": "pm*(1/2)", "fact_expr": "G: Line;Expression(G) = (y = k * (x + 1));C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number;Intersection(G, C) = {A, B};Focus(C) = F;F: Point;Abs(LineSegmentOf(F, A)) + Abs(LineSegmentOf(F, B)) = 16;k: Number", "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/dev/parabola/problem_0356.png", "answer": "pm*sqrt(15)", "fact_expr": "l: Line;C: Parabola;p: Number;Q: Point;F: Point;P: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = P;Intersection(l, Directrix(C)) = Q;VectorOf(Q, F) = 3*VectorOf(F, P)", "coords": {} }, { "index": 357, "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/dev/ellipse/problem_0357.png", "answer": "8/3", "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};Radius(InscribedCircle(TriangleOf(P, F1, F2))) = 1;Quadrant(P) = 1", "coords": {} }, { "index": 358, "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/dev/parabola/problem_0358.png", "answer": "2", "fact_expr": "C: Parabola;G: Line;A: Point;F: Point;B: Point;k:Number;Expression(C) = (y^2 = 4*x);k>0;Expression(G) = (y = k*(x - 2));Focus(C) = F;Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(B, F)) = 2/5;Negation(A=B)", "coords": {} }, { "index": 359, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0359.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;LeftFocus(C) = F;H: Line;PointOnCurve(F, H);Z1: Line;Z2: Line;Asymptote(C) = {Z1, Z2};Q: Point;Intersection(H, Z1) = Q;P: Point;Intersection(H, Z2) = P;2*VectorOf(F, Q) = VectorOf(F, P);O: Origin;G: Circle;Center(G) = O;IsTangent(G, LineOf(F, P));TangentPoint(G, LineOf(F, P)) = Q", "coords": {} }, { "index": 360, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 361, "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/dev/parabola/problem_0361.png", "answer": "y=3*x-2", "fact_expr": "G: Curve;Expression(G) = (y = x^2);H: Point;Coordinate(H) = (1, 1)", "coords": { "H": [ 1.0, 1.0 ] } }, { "index": 362, "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/dev/parabola/problem_0362.png", "answer": "9/2", "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 3.5, 4.0 ] } }, { "index": 363, "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/dev/ellipse/problem_0363.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 364, "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/dev/ellipse/problem_0364.png", "answer": "3*sqrt(2)/2", "fact_expr": "l: Line;C: Curve;P: Point;Expression(C) = (x^2/3 + y^2 = 1);Expression(l) = (x + y + 1 = 0);PointOnCurve(P, C)", "coords": {} }, { "index": 365, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0365.png", "answer": "(1, 3*sqrt(2)/4]", "fact_expr": "E: Hyperbola;a: Number;b: Number;C: Parabola;A: Point;P: Point;F: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (y^2 = 8*(a*x));RightVertex(E) = A;Focus(C) = F;PointOnCurve(P, Asymptote(E));IsPerpendicular(VectorOf(A,P),VectorOf(F,P))", "coords": {} }, { "index": 366, "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/dev/ellipse/problem_0366.png", "answer": "y^2=-8*x", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);P: Point;Distance(P, LeftFocus(G)) = Distance(P, H);H: Line;Expression(H) = (x = 2)", "coords": {} }, { "index": 367, "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/dev/ellipse/problem_0367.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;A: Point;B: Point;O: Origin;a > b;b > 0;LeftVertex(G)=A;RightVertex(G)=B;P:Point;PointOnCurve(P, G);Negation(P=A);Negation(P=B);Slope(LineOf(P, A)) *Slope(LineOf(P, B)) = -1/2", "coords": {} }, { "index": 368, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 369, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/parabola/problem_0369.png", "answer": "sqrt(55)", "fact_expr": "G: Parabola;M: Point;P0: Point;Expression(G) = (y^2 = 10*x);Coordinate(M) = (8, 0);PointOnCurve(P0, G)", "coords": { "M": [ 8.0, 0.0 ] } }, { "index": 370, "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/dev/hyperbola/problem_0370.png", "answer": "y^2=20*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*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": 371, "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/dev/parabola/problem_0371.png", "answer": "2", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;k:Number;Expression(C) = (y^2 = 8*x);Coordinate(M) = (-2, 2);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -2.0, 2.0 ] } }, { "index": 372, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.06247315928339958, "direction": "up" }, "optimization": { "final_loss": 0.0014067133888602257, "converged": true, "iterations": 338 }, "output_path": "results/dev/parabola/problem_0372.png", "answer": "y=-1/16", "fact_expr": "C: Parabola;G: Point;O: Origin;Coordinate(G) = (-1, 4);Vertex(C) = O;PointOnCurve(Focus(C), yAxis);PointOnCurve(G,C)", "coords": { "G": [ -1.0, 4.0 ] } }, { "index": 373, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0373.png", "answer": "(1,sqrt(7)-1)", "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,RightPart(C)) = True;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Circle;IsInscribedCircle(M,TriangleOf(P,F1,F2)) = True;A: Point;TangentPoint(LineSegmentOf(P, F1), M) = A;Abs(LineSegmentOf(P,A)) = c^2/2*a;c: Number;HalfFocalLength(C) = c;Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2)) > 2", "coords": {} }, { "index": 374, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0374.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, C) = {A, B};G: Circle;IsDiameter(LineSegmentOf(A, B), G);M: Point;Coordinate(M) = (-p/2, 3);TangentPoint(Directrix(C), G) = M;O: Origin;Area(TriangleOf(A, O, B)) = sqrt(13)", "coords": {} }, { "index": 375, "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/dev/hyperbola/problem_0375.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;A: Point;F: Point;B: Point;p>0;Focus(H)=F;l:Line;PointOnCurve(F,l);IsParallel(l,OneOf(Asymptote(G)));Intersection(l,H)={A,B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(A, F)) = 2", "coords": {} }, { "index": 376, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0376.png", "answer": "(sqrt(5)+1)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;L: Line;A: Point;B: Point;F: Point;O:Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, L);IsPerpendicular(L,xAxis);Intersection(L, G) = {A, B};IsDiameter(LineSegmentOf(A,B),H);PointOnCurve(O,H)", "coords": {} }, { "index": 377, "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/dev/parabola/problem_0377.png", "answer": "5/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;Distance(MidPoint(LineSegmentOf(A, B)), yAxis) = 1", "coords": {} }, { "index": 378, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0378.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;G: Parabola;p: Number;M: Point;A: Point;b>0;Expression(C) = (x^2 - y^2/b^2 = 1);p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (1, y0);y0:Number;PointOnCurve(M, G);Distance(M, Focus(G)) = 5;LeftVertex(C) = A;IsPerpendicular(OneOf(Asymptote(C)),LineOf(A,M))", "coords": {} }, { "index": 379, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 380, "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/dev/hyperbola/problem_0380.png", "answer": "6", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (5, 0);RightFocus(C)=F;Distance(F,Asymptote(C))=4", "coords": { "F": [ 5.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 381, "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/dev/parabola/problem_0381.png", "answer": "x^2+(y-1)^2=10", "fact_expr": "C: Circle;G: Parabola;Expression(G) = (x^2 = 4*y);Center(C) = Focus(G);H: Line;Expression(H) = (4*x - 3*y - 2 = 0);A: Point;B: Point;Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 382, "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/dev/hyperbola/problem_0382.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "C: Hyperbola;l: Line;OneOf(Asymptote(C)) = l;Expression(l) = (y = 2*x);Length(RealAxis(C)) < 4", "coords": {} }, { "index": 383, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0383.png", "answer": "sqrt(5)-1", "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, C) = True;IsDiameter(LineSegmentOf(F1, F2), G) = True;G: Circle;Intersection(G, Asymptote(C)) = Q;Q: Point;Quadrant(P) = 1;Quadrant(Q) = 1;MidPoint(LineSegmentOf(Q, F2)) = P", "coords": {} }, { "index": 384, "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/dev/hyperbola/problem_0384.png", "answer": "y^2 = 12*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);RightFocus(G) = Focus(H);H: Parabola", "coords": {} }, { "index": 385, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0385.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;D: Point;OneOf(Vertex(ImageinaryAxis(C))) = D;G: Line;Expression(G) = (x = 2*a);A: Point;B: Point;Intersection(G, C) = {A, B};PointOnCurve(Orthocenter(TriangleOf(A, B, D)), OneOf(Asymptote(C)))", "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/dev/hyperbola/problem_0386.png", "answer": "-1", "fact_expr": "C1: Hyperbola;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;LeftDirectrix(C1) = l;F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;C2: Parabola;Directrix(C2) = l;Focus(C2) = F2;M: Point;Intersection(C1, C2) = M", "coords": {} }, { "index": 387, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 6.708203932499369, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0387.png", "answer": "y=pm*x*sqrt(5)/2\n3/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/36 - y^2/45 = 1)", "coords": {} }, { "index": 388, "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/dev/ellipse/problem_0388.png", "answer": "sqrt(10)/5", "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 389, "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/dev/parabola/problem_0389.png", "answer": "x+y-1=0", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);L: Line;PointOnCurve(Focus(G), L);H: Circle;Expression(H) = (2*y - 4*x + x^2 + y^2 = 0);WhenMax(Length(InterceptChord(L, H)))", "coords": {} }, { "index": 390, "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/dev/hyperbola/problem_0390.png", "answer": "32/5", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P,RightFocus(G)) = 8", "coords": {} }, { "index": 391, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 392, "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/dev/parabola/problem_0392.png", "answer": "{1, 3}", "fact_expr": "L: Line;Expression(L) = (y = -t + x);t: Number;C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;Intersection(L, C) = {A, B};Negation(A=B);O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = -3", "coords": {} }, { "index": 393, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 394, "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/dev/ellipse/problem_0394.png", "answer": "{(x^2/8+y^2/6=1),(y^2/25/3+x^2/25/4=1)}", "fact_expr": "G1: Ellipse;Expression(G1) = (x^2/4 + y^2/3 = 1);G2: Ellipse;Eccentricity(G1) = Eccentricity(G2);H: Point;Coordinate(H) = (2, -sqrt(3));PointOnCurve(H,G2) = True", "coords": { "H": [ 2.0, -1.7320508075688772 ] } }, { "index": 395, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 396, "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/dev/parabola/problem_0396.png", "answer": "(4*sqrt(2),4*sqrt(6))", "fact_expr": "l: Line;G: Parabola;O: Point;A: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 0);Vertex(G)=O;Inclination(l)=pi/4;PointOnCurve(O,l);PointOnCurve(A,l);IsIntersect(LineSegmentOf(O,A),l);Intersection(l,LineSegmentOf(O,A))={P,Q}", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 397, "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/dev/hyperbola/problem_0397.png", "answer": "3*sqrt(5)+13", "fact_expr": "F: Point;C: Hyperbola;Expression(C) = (x^2/16 - y^2/9 = 1);RightFocus(C) = F;P: Point;PointOnCurve(P, LeftPart(C)) = True;A: Point;Coordinate(A) = (1, 3)", "coords": { "A": [ 1.0, 3.0 ] } }, { "index": 398, "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/dev/parabola/problem_0398.png", "answer": "2*sqrt(3)", "fact_expr": "l: Line;C: Parabola;G: Line;B: Point;A: Point;O: Origin;Expression(C) = (y^2 = x);Expression(G) = (y = 1);Intersection(l, C) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A, B)), G);DotProduct(VectorOf(O, A), VectorOf(O, B)) = 2", "coords": {} }, { "index": 399, "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/dev/hyperbola/problem_0399.png", "answer": "(-1/2, pm*3/8)", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*x);G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 400, "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/dev/ellipse/problem_0400.png", "answer": "17/4", "fact_expr": "A: Point;Coordinate(A) = (4, 0);B: Point;Coordinate(B) = (2, 2);M: Point;PointOnCurve(M, G) = True;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)", "coords": { "A": [ 4.0, 0.0 ], "B": [ 2.0, 2.0 ] } }, { "index": 401, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0401.png", "answer": "sqrt(7)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);Quadrant(P) = 1;PointOnCurve(P, C);l: Line;TangentOnPoint(P, C) = l;Distance(F1, l)*Distance(F2, l) = 3", "coords": {} }, { "index": 402, "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/dev/parabola/problem_0402.png", "answer": "y=-3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y)", "coords": {} }, { "index": 403, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0403.png", "answer": "(0, pm*1)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/3 = 1)", "coords": {} }, { "index": 404, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0404.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": 405, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0405.png", "answer": "[0,b^4/c^2]", "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;M: Point;O: Origin;x0:Number;y0:Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (x0, y0);PointOnCurve(M, C);HalfFocalLength(C) = c;Abs(LineSegmentOf(M, O)) <= c", "coords": {} }, { "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/dev/hyperbola/problem_0406.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;C: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(G)=A;Slope(H)=-1;PointOnCurve(A,H);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=B;Intersection(H,L2)=C;VectorOf(A, B) = VectorOf(B, C)/2", "coords": {} }, { "index": 407, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 408, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 409, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 14.48876953125, "direction": "right" }, "optimization": { "final_loss": 22.99686622619629, "converged": false, "iterations": 500 }, "output_path": "results/dev/parabola/problem_0409.png", "answer": "y^2=(-9/2)*x", "fact_expr": "C: Parabola;G: Point;O: Origin;Coordinate(G) = (-2, 3);Vertex(C) = O;PointOnCurve(Focus(C), xAxis);PointOnCurve(G, C)", "coords": { "G": [ -2.0, 3.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 410, "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/dev/parabola/problem_0410.png", "answer": "pi/2", "fact_expr": "l: Line;G: Parabola;P: Point;Q: Point;F: Point;Expression(G) = (y = -x^2/4);Coordinate(P) = (-4, -4);Focus(G) = F;TangentPoint(l, G) = P;Intersection(l, xAxis) = Q", "coords": { "P": [ -4.0, -4.0 ] } }, { "index": 411, "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/dev/hyperbola/problem_0411.png", "answer": "sqrt(5)/5", "fact_expr": "C: Hyperbola;A: Point;P: Point;Q: Point;B: Point;T: Curve;M: Point;Expression(C) = (x^2/4 - y^2/5 = 1);LeftVertex(C) = A;RightVertex(C) = B;IsChordOf(LineSegmentOf(P, Q), C);IsPerpendicular(LineSegmentOf(P, Q), xAxis);Intersection(LineOf(P, A), LineOf(Q, B)) = M;Locus(M)=T", "coords": {} }, { "index": 412, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0412.png", "answer": "11", "fact_expr": "l: Line;G: Hyperbola;C: Parabola;A: Point;B: Point;P: Point;F:Point;O: Origin;Expression(G) = (x^2/3 - y^2/6 = 1);Coordinate(P) = (2, 0);Vertex(C) = O;Focus(C) =F;RightFocus(G)=F;PointOnCurve(P, l);Slope(l) = 1;Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A,B),C)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 413, "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/dev/ellipse/problem_0413.png", "answer": "2/11", "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;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2));Abs(LineSegmentOf(A, F2)) = 2*Abs(LineSegmentOf(A, F1))", "coords": {} }, { "index": 414, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0414.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Parabola;p: Number;H: Circle;A: Point;C: Point;D:Point;B: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2 + y^2 = 1);Intersection(H, G) = {A, B};Intersection(H,Directrix(G))={C,D};MidPoint(LineSegmentOf(A, C)) = O", "coords": {} }, { "index": 415, "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/dev/parabola/problem_0415.png", "answer": "4/5", "fact_expr": "l: Line;Expression(l) = (4*x - 3*y + 6 = 0);G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;PointOnCurve(P, G);Distance(P, l) = d;d: Number", "coords": {} }, { "index": 416, "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/dev/hyperbola/problem_0416.png", "answer": "12", "fact_expr": "G: Hyperbola;H: Parabola;m: Real;Expression(G) = (x^2/6 - y^2/3 = 1);Expression(H) = (y^2 = m*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 417, "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/dev/ellipse/problem_0417.png", "answer": "2-sqrt(3)", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;M: Point;F1: 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(F2,H);Inclination(H)=ApplyUnit(120,degree);OneOf(Intersection(H,G))=M;IsPerpendicular(LineSegmentOf(M,F1),xAxis)", "coords": {} }, { "index": 418, "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/dev/parabola/problem_0418.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);IsPerpendicular(LineSegmentOf(A,B),xAxis);Abs(LineSegmentOf(A, B)) = 4*sqrt(3)", "coords": {} }, { "index": 419, "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/dev/hyperbola/problem_0419.png", "answer": "3", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2 = 2);Coordinate(A) = (0, 2);RightFocus(C) = F;PointOnCurve(P, LeftPart(C));WhenMin(Perimeter(TriangleOf(A, P, F)))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 420, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0420.png", "answer": "25/4", "fact_expr": "l: Line;G: Parabola;p: Number;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Slope(l)=4/3;Focus(G) = F;PointOnCurve(F,l);Intersection(l, G) = {A, B}", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 421, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0421.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": 422, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 423, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 424, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 425, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 426, "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/dev/hyperbola/problem_0426.png", "answer": "16", "fact_expr": "m: Number;F: Point;Coordinate(F) = (0, 5);G: Hyperbola;Expression(G) = (-x^2/9 + y^2/m = 1);OneOf(Focus(G)) = F", "coords": { "F": [ 0.0, 5.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 427, "success": false, "conic_type": "circle", "error": "validation: circle_point_off_curve", "params": { "center": [ 3.0, 0.0 ], "radius": 10.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0427.png", "answer": "x^2/25+y^2/16=1", "fact_expr": "M: Circle;A: Point;Coordinate(A) = (-3, 0);PointOnCurve(A, M);Expression(B)=((x-3)^2+y^2=100);M1:Point;Center(M)=M1;IsInTangent(M,B);B:Circle", "coords": { "A": [ -3.0, 0.0 ] }, "validation_reasons": [ "circle_point_off_curve" ] }, { "index": 428, "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/dev/parabola/problem_0428.png", "answer": "5", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (4, y0);y0:Number;PointOnCurve(M, G);Focus(G)=F;F: Point", "coords": {} }, { "index": 429, "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/dev/parabola/problem_0429.png", "answer": "(1,2)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 2);PointOnCurve(P, G);Focus(G) = F;WhenMin(LineSegmentOf(P,A)+LineSegmentOf(P,F))", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 430, "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/dev/ellipse/problem_0430.png", "answer": "x^2/3-y^2/2=1", "fact_expr": "G: Hyperbola;H: Ellipse;A: Point;Expression(H) = (x^2/9 + y^2/4 = 1);Coordinate(A) = (3,-2);PointOnCurve(A, G);Focus(G)=Focus(H)", "coords": { "A": [ 3.0, -2.0 ] } }, { "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/dev/hyperbola/problem_0431.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a: Number;a>0;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 432, "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/dev/hyperbola/problem_0432.png", "answer": "4 + 3*sqrt(5)", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 6);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 6.0 ] } }, { "index": 433, "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/dev/hyperbola/problem_0433.png", "answer": "y=pm*2*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/16 = 1)", "coords": {} }, { "index": 434, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0434.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;P: Point;PointOnCurve(P, RightPart(G));F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;I: Point;Incenter(TriangleOf(P, F1, F2)) = I;TriangleOf(I, P, F1) = (1/2)*TriangleOf(I, F1, F2) + TriangleOf(I, P, F2)", "coords": {} }, { "index": 435, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 436, "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/dev/hyperbola/problem_0436.png", "answer": "3/16", "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;Negation(m*n=0);Expression(G) = (-y^2/n + x^2/m = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 437, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 438, "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/dev/ellipse/problem_0438.png", "answer": "12*sqrt(2)/7", "fact_expr": "G: Ellipse;H: Line;k: Number;F: Point;A: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y = k*x - 1);LeftFocus(G) = F;Intersection(H, G) = {A, B};WhenMax(Perimeter(TriangleOf(F,A,B)))", "coords": {} }, { "index": 439, "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/dev/hyperbola/problem_0439.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/4 = 1)", "coords": {} }, { "index": 440, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 441, "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/dev/circle/problem_0441.png", "answer": "x^2/25+y^2/16=1", "fact_expr": "G: Circle;IsOutTangent(G,Q1) = True;Q1: Circle;Expression(Q1) = ((x+3)^2+y^2=1);Q2: Circle;Expression(Q2) = ((x-3)^2+y^2=81);IsInTangent(G,Q2) = True", "coords": {} }, { "index": 442, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 443, "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/dev/parabola/problem_0443.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": 444, "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/dev/ellipse/problem_0444.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;A: Point;P: Point;Expression(C) = (x^2/4 + y^2 = 1);Coordinate(A) = (1, 0);PointOnCurve(P,C)", "coords": { "A": [ 1.0, 0.0 ] } }, { "index": 445, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0445.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;e: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G),xAxis);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(RightDirectrix(G), L1) = A;Intersection(RightDirectrix(G), L2) = B;RightFocus(G) = F;DotProduct(VectorOf(F, A), VectorOf(F, B)) = 0;Eccentricity(G) = e", "coords": {} }, { "index": 446, "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/dev/hyperbola/problem_0446.png", "answer": "{2, (2*sqrt(15))/5}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);Slope(H)>0;A: Point;B: Point;Intersection(H, Asymptote(G)) = {A, B};O: Origin;Area(TriangleOf(O, A, B)) = 16*sqrt(3)", "coords": {} }, { "index": 447, "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/dev/ellipse/problem_0447.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;c: Number;F: Point;RightFocus(C) = F;Coordinate(F) = (c, 0);P: Point;Q: Point;PointOnCurve(P, C);PointOnCurve(Q, C);M: Point;Coordinate(M) = (-c/2, 0);Distance(M, LineOf(F, P)) = c/2;Incenter(TriangleOf(P, Q, F)) = M;e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 448, "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/dev/ellipse/problem_0448.png", "answer": "[(sqrt(17)-3)/4, 1)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);e: Number;Eccentricity(C) = e;LineSegmentOf(P, F1) = (2*e)*LineSegmentOf(P, F2)", "coords": {} }, { "index": 449, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 450, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 451, "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/dev/hyperbola/problem_0451.png", "answer": "2", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Sin(AngleOf(P, F2, F1))/Sin(AngleOf(P, F1, F2)) = e;e:Number;Eccentricity(G)=e", "coords": {} }, { "index": 452, "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/dev/hyperbola/problem_0452.png", "answer": "pm*sqrt(2)", "fact_expr": "G: Hyperbola;H: Circle;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 + (-a + x)^2 = 1);IsTangent(H,Asymptote(G))", "coords": {} }, { "index": 453, "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/dev/ellipse/problem_0453.png", "answer": "4*sqrt(7)/7", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {A,B}", "coords": {} }, { "index": 454, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 455, "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/dev/parabola/problem_0455.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Slope(H) = 1;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 456, "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/dev/parabola/problem_0456.png", "answer": "5", "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": 457, "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/dev/parabola/problem_0457.png", "answer": "{(x-1)^2+(y-2)^2=4,(x-1)^2+(y+2)^2=4}", "fact_expr": "G: Parabola;C: Circle;Expression(G) = (y^2 = 4*x);PointOnCurve(Center(C),G);IsTangent(C,xAxis);IsTangent(C,Directrix(G))", "coords": {} }, { "index": 458, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0458.png", "answer": "4*a+2*m", "fact_expr": "G: Hyperbola;a: Number;b: Number;A: Point;B: Point;F1:Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;Focus(G)={F1,F2};Abs(LineSegmentOf(A, B)) = m;IsChordOf(LineSegmentOf(A, B),LeftPart(G));PointOnCurve(F1,LineSegmentOf(A, B));m:Number", "coords": {} }, { "index": 459, "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/dev/ellipse/problem_0459.png", "answer": "(2*sqrt(5))/5", "fact_expr": "H: Line;Expression(H) = (x + 2*y - 2 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Focus(G)), H);PointOnCurve(OneOf(Vertex(G)), H)", "coords": {} }, { "index": 460, "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/dev/hyperbola/problem_0460.png", "answer": "{14, 2}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 8", "coords": {} }, { "index": 461, "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/dev/parabola/problem_0461.png", "answer": "10", "fact_expr": "l: Line;Slope(l) = 2;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B}", "coords": {} }, { "index": 462, "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/dev/hyperbola/problem_0462.png", "answer": "16/5", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 463, "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/dev/hyperbola/problem_0463.png", "answer": "(pm*sqrt(5), 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/2 = 1)", "coords": {} }, { "index": 464, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0464.png", "answer": "x=-3/2", "fact_expr": "l: Line;C: Parabola;P: Number;P>0;Expression(l)=(y = x + 3/2);Expression(C) = (y^2 = 2*(P*x));NumIntersection(l,C)=1", "coords": {} }, { "index": 465, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0465.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/9 = 1)", "coords": {} }, { "index": 466, "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/dev/ellipse/problem_0466.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G),H);PointOnCurve(Endpoint(MinorAxis(G)),H)", "coords": {} }, { "index": 467, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0467.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Distance(LeftVertex(G), Focus(H)) = 4;Coordinate(Intersection(OneOf(Asymptote(G)),Directrix(H)))=(-2,-1)", "coords": {} }, { "index": 468, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.015625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/parabola/problem_0468.png", "answer": "(0, 1/64)", "fact_expr": "G: Parabola;Expression(G) = (y = 16*x^2)", "coords": {} }, { "index": 469, "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/dev/ellipse/problem_0469.png", "answer": "4*sqrt(3)", "fact_expr": "C: Ellipse;P: Point;Q: Point;F2: Point;Expression(C) = (x^2/3 + y^2/2 = 1);RightFocus(C) = F2;PointOnCurve(P, C);PointOnCurve(Q, C);LineSegmentOf(P, Q) = 2*sqrt(2);WhenMax(LineSegmentOf(P,F2)+LineSegmentOf(Q,F2))", "coords": {} }, { "index": 470, "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/dev/parabola/problem_0470.png", "answer": "16/3", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = pi/3;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)", "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/dev/parabola/problem_0471.png", "answer": "8", "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;B: Point;F: Point;N: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 0);PointOnCurve(M, l);Intersection(l, C) = {A, B};Between(A, M, B);Focus(C) = F;VectorOf(N, F) = 6*VectorOf(A, F)", "coords": { "M": [ -1.0, 0.0 ] } }, { "index": 472, "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/dev/ellipse/problem_0472.png", "answer": "14", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/100 + y^2/36 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 473, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 474, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0474.png", "answer": "y=pm*(4/3)*x", "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));Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = (3/5)*Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 475, "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/dev/ellipse/problem_0475.png", "answer": "16/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 476, "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/dev/parabola/problem_0476.png", "answer": "3", "fact_expr": "C: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),C))=Q;VectorOf(F, P) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 477, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 478, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 479, "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/dev/hyperbola/problem_0479.png", "answer": "4/5", "fact_expr": "l: Line;G: Hyperbola;Q: Point;P: Point;A: Point;Expression(G) = (x^2/5 - y^2/4 = 1);Coordinate(A) = (1, 1);Intersection(l, G) = {P, Q};MidPoint(LineSegmentOf(P,Q))=A", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 480, "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/dev/hyperbola/problem_0480.png", "answer": "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)", "coords": {} }, { "index": 481, "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/dev/parabola/problem_0481.png", "answer": "16", "fact_expr": "m: Line;E: Parabola;A: Point;M: Point;F: Point;N: Point;B: Point;l: Line;Expression(E) = (y^2 = 12*x);Focus(E) = F;Directrix(E) = l;PointOnCurve(F,m);Intersection(m, E) = {A, B};PointOnCurve(A,LineSegmentOf(A,M));IsPerpendicular(LineSegmentOf(A,M),l);FootPoint(LineSegmentOf(A,M),l)=M;MidPoint(LineSegmentOf(A, M)) = N;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(F, N))", "coords": {} }, { "index": 482, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 0.0 ], "radius": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0482.png", "answer": "x^2/9+y^2/5=1", "fact_expr": "G: Circle;A: Point;N: Point;M: Point;P: Point;Expression(G) = (y^2 + (x + 2)^2 = 36);Coordinate(N) = (2, 0);Center(G)=M;PointOnCurve(A, G);Intersection(PerpendicularBisector(LineSegmentOf(A,N)), LineSegmentOf(M, A)) = P", "coords": {} }, { "index": 483, "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/dev/hyperbola/problem_0483.png", "answer": "24", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/24 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);3*LineSegmentOf(P, F1) = 4*LineSegmentOf(P, F2)", "coords": {} }, { "index": 484, "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/dev/hyperbola/problem_0484.png", "answer": "x^2/2 - y^2/8 = 1", "fact_expr": "C: Hyperbola;Expression(OneOf(Asymptote(C))) = (y = 2*x)", "coords": {} }, { "index": 485, "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/dev/ellipse/problem_0485.png", "answer": "x^2-y^2=2", "fact_expr": "H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);G: Hyperbola;Focus(H) = Focus(G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};IsPerpendicular(L1, L2)", "coords": {} }, { "index": 486, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 487, "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/dev/hyperbola/problem_0487.png", "answer": "9", "fact_expr": "G: Hyperbola;H: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9-y^2/16 = 1);Expression(C) = (y^2 + (x + 5)^2 = 4);PointOnCurve(P,RightPart(G));PointOnCurve(M,C);PointOnCurve(N,H);C:Circle;Expression(H)=(y^2 + (x-5)^2 = 1)", "coords": {} }, { "index": 488, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0488.png", "answer": "sqrt(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);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(VectorOf(P, F1) + VectorOf(P, F2)) = sqrt(Abs(VectorOf(P, F1))^2 + Abs(VectorOf(P, F2))^2);Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 489, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 490, "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/dev/ellipse/problem_0490.png", "answer": "y=-x+3", "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/8 + y^2/4 = 1);Coordinate(MidPoint(H)) = (2, 1);IsChordOf(H,G)", "coords": {} }, { "index": 491, "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/dev/ellipse/problem_0491.png", "answer": "16", "fact_expr": "G: Ellipse;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 492, "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/dev/parabola/problem_0492.png", "answer": "x-y+1=0", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);PointOnCurve(P, G);Quadrant(P)=1;Focus(G) = F;WhenMin(Abs(LineSegmentOf(P, F))/Abs(LineSegmentOf(P, A)))", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 493, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/parabola/problem_0494.png", "answer": "4", "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)) = 4/3", "coords": {} }, { "index": 495, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 496, "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/dev/hyperbola/problem_0496.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)*a", "coords": {} }, { "index": 497, "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/dev/parabola/problem_0497.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);F: Point;Focus(G) = F;l: Line;Directrix(G) = l", "coords": {} }, { "index": 498, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 499, "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/dev/ellipse/problem_0499.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4*a);a: Real;Negation(a=0);H: Ellipse;Expression(H) = (x^2/16 + y^2/9 = 1);RightFocus(G) = OneOf(Vertex(H))", "coords": {} }, { "index": 500, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 501, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 502, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0502.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 503, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 0.9927864074707031, "direction": "up" }, "optimization": { "final_loss": 0.00015326803259085864, "converged": true, "iterations": 44 }, "output_path": "results/dev/parabola/problem_0503.png", "answer": "5", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Coordinate(P) = (4, 4);Focus(C) = F;PointOnCurve(P, C)", "coords": { "P": [ 4.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 504, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 505, "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/dev/ellipse/problem_0505.png", "answer": "x^2+x+3*y^2=0", "fact_expr": "C: Ellipse;G: Point;A: Point;B: Point;Expression(C) = (x^2/3 + y^2 = 1);Coordinate(G) = (-1, 0);IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(G,LineSegmentOf(A,B))", "coords": { "G": [ -1.0, 0.0 ] } }, { "index": 506, "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/dev/hyperbola/problem_0506.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1)", "coords": {} }, { "index": 507, "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/dev/ellipse/problem_0507.png", "answer": "3*sqrt(7)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/25 = 1);M: Point;PointOnCurve(M, G) = True;T: Point;Coordinate(T) = (0, -3);N: Point;Abs(LineSegmentOf(N, T)) = 1;AngleOf(M, N, T) = ApplyUnit(90, degree)", "coords": { "T": [ 0.0, -3.0 ] } }, { "index": 508, "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/dev/parabola/problem_0508.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, F)))", "coords": { "A": [ -1.0, 0.0 ] } }, { "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/dev/ellipse/problem_0509.png", "answer": "(-oo,-1]+[1,+oo)", "fact_expr": "C: Ellipse;G: Circle;A: Point;B: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Expression(G) = (x^2 + y^2 = 1);PointOnCurve(A,C);PointOnCurve(B,C);PointOnCurve(MidPoint(LineSegmentOf(A,B)), G)", "coords": {} }, { "index": 510, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 511, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0511.png", "answer": "2*sqrt(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;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, LeftPart(G));IsParallel(LineOf(A, F1), H);H: Line;Expression(H) = (y = x*(b/a));Perimeter(TriangleOf(A, F1, F2)) = 8*a", "coords": {} }, { "index": 512, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0512.png", "answer": "12", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;Expression(G) = (x^2/(2*p) - y^2/p = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 513, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 514, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0514.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/2 + x^2/a = 1);Eccentricity(G) = sqrt(3*a)", "coords": {} }, { "index": 515, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 516, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 517, "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/dev/ellipse/problem_0517.png", "answer": "{9/2, 8}", "fact_expr": "G: Ellipse;Expression(G) = (y^2/6 + x^2/m = 1);m: Number;Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 518, "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/dev/parabola/problem_0518.png", "answer": "16/3", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 519, "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/dev/ellipse/problem_0519.png", "answer": "4", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/9 + y^2/3 = 1);H: Line;PointOnCurve(F1, H);Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 8", "coords": {} }, { "index": 520, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0520.png", "answer": "sqrt(5), sqrt(5)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B: Point;a>0;b>0;L1:Line;L2:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x=4*a);Asymptote(C)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;Area(TriangleOf(O,A,B))=32;FocalLength(C) = 2*sqrt(5)", "coords": {} }, { "index": 521, "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/dev/ellipse/problem_0521.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;n: Number;m: Number;m>0;n>0;2/n + 1/m = 1;WhenMin(m*n);Expression(G) = (y^2/n^2 + x^2/m^2 = 1)", "coords": {} }, { "index": 522, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0522.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^2/b", "coords": {} }, { "index": 523, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 524, "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/dev/ellipse/problem_0524.png", "answer": "12*(2+sqrt(2))/5\n12*(2-sqrt(2))/5", "fact_expr": "G: Ellipse;H: Line;P: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Expression(H) = (3*x - 4*y = 24);PointOnCurve(P, 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/dev/hyperbola/problem_0525.png", "answer": "(1, 3/2)", "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) = (-6*y + x^2 + y^2 + 5 = 0);NumIntersection(Asymptote(G), H) = 0", "coords": {} }, { "index": 526, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 527, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0527.png", "answer": "2", "fact_expr": "A: Point;B: Point;G: Parabola;p: Number;H: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F, H);Inclination(H)=ApplyUnit(45, degree);Intersection(H,G) = {B, A};Length(LineSegmentOf(A, B)) = 8;F: Point", "coords": {} }, { "index": 528, "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/dev/hyperbola/problem_0528.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (y^2/4 - x^2/b^2 = 1);FocalLength(G) = 8", "coords": {} }, { "index": 529, "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/dev/ellipse/problem_0529.png", "answer": "sqrt(6)", "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;M: Point;PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(60, degree);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 4", "coords": {} }, { "index": 530, "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/dev/ellipse/problem_0530.png", "answer": "32/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F2: Point;RightFocus(G) = F2;H: LineSegment;IsChordOf(H, G);PointOnCurve(F2, H)", "coords": {} }, { "index": 531, "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/dev/hyperbola/problem_0531.png", "answer": "y=pm*(2/3)*x", "fact_expr": "G: Hyperbola;b: Number;H: Point;Expression(G) = (x^2/9 - y^2/b^2 = 1);Coordinate(H) = (sqrt(13), 0);RightFocus(G) = H", "coords": { "H": [ 3.605551275463989, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 532, "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/dev/parabola/problem_0532.png", "answer": "x^2-y^2/3=1", "fact_expr": "O: Origin;G:Hyperbola;H: Parabola;Expression(H) = (y^2 = -8*x);Eccentricity(G)=2;Center(G)=O;OneOf(Vertex(G))=Focus(H)", "coords": {} }, { "index": 533, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0533.png", "answer": "(2,+oo)", "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);XCoordinate(P)=3*a/2;Distance(P,RightFocus(G))>Distance(P,LeftDirectrix(G))", "coords": {} }, { "index": 534, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0534.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;FocalLength(C) = 4*sqrt(5);Slope(OneOf(Asymptote(C))) = 2", "coords": {} }, { "index": 535, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 536, "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/dev/parabola/problem_0536.png", "answer": "9/2", "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 3.5, 4.0 ] } }, { "index": 537, "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/dev/hyperbola/problem_0537.png", "answer": "-4", "fact_expr": "C: Hyperbola;A: Point;B: Point;P: Point;Expression(C) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (-5, 0);Coordinate(B) = (5, 4);PointOnCurve(P, RightPart(C))", "coords": { "A": [ -5.0, 0.0 ], "B": [ 5.0, 4.0 ] } }, { "index": 538, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0538.png", "answer": "(-oo, -2*sqrt(2)] + (0, +oo)", "fact_expr": "l: Line;G: Parabola;p: Number;O: Origin;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};PointOnCurve(A, xAxis);Abs(VectorOf(O, A)) >= Abs(VectorOf(F, A))", "coords": {} }, { "index": 539, "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/dev/hyperbola/problem_0539.png", "answer": "x^2/(9/4)-y^2/4=1", "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (3, -2*sqrt(3));Asymptote(C) = Asymptote(G);PointOnCurve(A, C)", "coords": { "A": [ 3.0, -3.4641016151377544 ] } }, { "index": 540, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.2152259349823, "b": 1.4676733384680494, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.580150581040369e-05, "converged": true, "iterations": 89 }, "output_path": "results/dev/ellipse/problem_0540.png", "answer": "x^2/16+y^2/7=1", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;Eccentricity(C)=3/4;PointOnCurve(P, C);Area(TriangleOf(F1, P, F2)) = 7;Radius(InscribedCircle(TriangleOf(F1,P,F2)))=1", "coords": {} }, { "index": 541, "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/dev/parabola/problem_0541.png", "answer": "4", "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 4*x);Focus(G) = F;M: Point;PointOnCurve(M, G);P: Point;Coordinate(P) = (3, 1)", "coords": { "P": [ 3.0, 1.0 ] } }, { "index": 542, "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/dev/hyperbola/problem_0542.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);Coordinate(Focus(G))=(4,0)", "coords": {} }, { "index": 543, "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/dev/parabola/problem_0543.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y)", "coords": {} }, { "index": 544, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 545, "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/dev/hyperbola/problem_0545.png", "answer": "3", "fact_expr": "C: Hyperbola;G: Circle;Expression(C) = (x^2/16 - y^2/9 = 1);RightFocus(C)=Center(G);IsTangent(G, Asymptote(C))", "coords": {} }, { "index": 546, "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/dev/ellipse/problem_0546.png", "answer": "sqrt(3)-1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;O: Origin;A: Point;B:Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*x);RightFocus(C) = F;Intersection(G,C)={A,B};Quadrant(A)=1;LineSegmentOf(O, A) = LineSegmentOf(O, F)", "coords": {} }, { "index": 547, "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/dev/ellipse/problem_0547.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 548, "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/dev/ellipse/problem_0548.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);WhenMin(1/Abs(LineSegmentOf(P, F1))+4/Abs(LineSegmentOf(P, F2)))", "coords": {} }, { "index": 549, "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/dev/hyperbola/problem_0549.png", "answer": "(3*x^2/2-y^2/2 = 1)", "fact_expr": "C: Hyperbola;C1: Point;Coordinate(C1) = (1,1);Expression(OneOf(Asymptote(C))) = (y = sqrt(3)*x)", "coords": { "C1": [ 1.0, 1.0 ] } }, { "index": 550, "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/dev/ellipse/problem_0550.png", "answer": "x^2/9 - y^2/27 = 1", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);G: Hyperbola;Focus(H) = Vertex(G);Eccentricity(G) = 2", "coords": {} }, { "index": 551, "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/dev/parabola/problem_0551.png", "answer": "4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Focus(C) = F;F: Point;A: Point;Coordinate(A) = (x0, y0);x0: Number;y0: Number;PointOnCurve(A, C) = True;Abs(LineSegmentOf(A, F)) = (5/4)*x0", "coords": {} }, { "index": 552, "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/dev/parabola/problem_0552.png", "answer": "(4,pm*4*sqrt(2))", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 6", "coords": {} }, { "index": 553, "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/dev/ellipse/problem_0553.png", "answer": "sqrt(3)/2", "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);Max(AngleOf(F1, P, F2)) = (2*pi)/3", "coords": {} }, { "index": 554, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0554.png", "answer": "sqrt(3)+1", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;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;Expression(l)=(y = sqrt(3)*x) ;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 555, "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/dev/ellipse/problem_0555.png", "answer": "(pm*2*sqrt(2), 0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1)", "coords": {} }, { "index": 556, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0556.png", "answer": "y=pm*(5/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/25 = 1)", "coords": {} }, { "index": 557, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 558, "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/dev/hyperbola/problem_0558.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Real;PointOnCurve(Intersection(RightDirectrix(G), Asymptote(G)), H)", "coords": {} }, { "index": 559, "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/dev/parabola/problem_0559.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;E: Ellipse;A: Point;B: Point;O:Origin;Center(E)=O;Eccentricity(E)=sqrt(3)/2;Expression(C) = (y^2 = 12*x);RightFocus(E) = Focus(C);Intersection(Directrix(C),E)={A,B}", "coords": {} }, { "index": 560, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0560.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, G);XCoordinate(A) = 3;Distance(A, Focus(G)) = 5", "coords": {} }, { "index": 561, "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/dev/ellipse/problem_0561.png", "answer": "{3, 16/3}", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 562, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 563, "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/dev/hyperbola/problem_0563.png", "answer": "{10,2}", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 6", "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/dev/ellipse/problem_0564.png", "answer": "x - 2*y + 4 = 0", "fact_expr": "E: Ellipse;b: Number;a: Number;M: Circle;r: Number;B: Point;A: Point;c: Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Expression(M) = ((x + 2)^2 + (y - 1)^2 = r^2);HalfFocalLength(E) = c;c = sqrt(3)*b;PointOnCurve(A, E);PointOnCurve(B, E);OneOf(Diameter(LineSegmentOf(A, B), M))", "coords": {} }, { "index": 565, "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/dev/ellipse/problem_0565.png", "answer": "x^2/4 + y^2/3 = 1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = 1/2;F2: Point;RightFocus(C) = F2;M: Point;G: Circle;Expression(G) = (x^2 + y^2 = b^2);PointOnCurve(M, G) = True;Quadrant(M) = 1;P: Point;Q: Point;l: Line;TangentOnPoint(M,G) = l;Perimeter(TriangleOf(P, F2, Q)) = 4;Intersection(l,C) = {P,Q}", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 566, "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/dev/ellipse/problem_0566.png", "answer": "[sqrt(2)/2, 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);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 567, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 568, "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/dev/ellipse/problem_0568.png", "answer": "5", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);Focus(G)={F1,F2}", "coords": {} }, { "index": 569, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0569.png", "answer": "4", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p > 0;F: Point;Focus(C) = F;P: Point;Coordinate(P) = (4, y0);y0: Number;PointOnCurve(P, C) = True;Abs(LineSegmentOf(F, P)) = 4", "coords": {} }, { "index": 570, "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/dev/ellipse/problem_0570.png", "answer": "(pm*1, 0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 571, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 572, "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/dev/ellipse/problem_0572.png", "answer": "4*sqrt(5)/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);H: Line;Expression(H) = (x - 2*y - 12 = 0);D: Point;PointOnCurve(D, G) = True", "coords": {} }, { "index": 573, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 574, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.872983346207417, "major_axis": "x", "x_coef": 16.0, "y_coef": 15.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0574.png", "answer": "21", "fact_expr": "G: Ellipse;N: Circle;E: Point;F: Point;P: Point;Expression(G) = (x^2/16 + y^2/15 = 1);Expression(N) = (y^2 + (x - 1)^2 = 4);PointOnCurve(P, G);IsDiameter(LineSegmentOf(E,F),N)", "coords": {} }, { "index": 575, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.449489742783178, "major_axis": "x", "x_coef": 8.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0575.png", "answer": "3/4", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/8 + y^2/6 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(-1,1)", "coords": {} }, { "index": 576, "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/dev/ellipse/problem_0576.png", "answer": "sqrt(2)", "fact_expr": "A: Point;F: Point;C: Ellipse;B: Point;l: Line;Expression(C) = (x^2/2 + y^2 = 1);RightFocus(C) = F;RightDirectrix(C) = l;In(A, l) = True;Intersection(LineSegmentOf(A, F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 577, "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/dev/parabola/problem_0577.png", "answer": "sqrt(2)/4", "fact_expr": "C: Parabola;P: Point;O: Origin;F: Point;x0: Number;y0: Number;Expression(C) = (y^2 = 2*x);Coordinate(P) = (x0, y0);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = (3/2)*x0", "coords": {} }, { "index": 578, "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/dev/hyperbola/problem_0578.png", "answer": "{(x^2/3-y^2/3=1),(x^2/8+y^2/2=1)}", "fact_expr": "G: Hyperbola;H: ConicSection;P:Point;Expression(G) = (x^2/4 - y^2/2 = 1);Coordinate(P) = (2, 1);Focus(G)=Focus(H);PointOnCurve(P,H)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 579, "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/dev/parabola/problem_0579.png", "answer": "2*(sqrt(2)-1)", "fact_expr": "G: Parabola;H: Line;M: Point;F: Point;N: Point;A: Point;B: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(F,H);Intersection(H,G) = {A,B};VectorOf(A, F) = VectorOf(F, B);Projection(A,l)=M;Projection(B,l)=N", "coords": {} }, { "index": 580, "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/dev/hyperbola/problem_0580.png", "answer": "y=pm*(2/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)", "coords": {} }, { "index": 581, "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/dev/parabola/problem_0581.png", "answer": "0", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);P: Point;Coordinate(P) = (0, -1);TangentOfPoint(P, C) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), C) = A;TangentPoint(LineOf(P, B), C) = B;A: Point;B: Point", "coords": { "P": [ 0.0, -1.0 ] } }, { "index": 582, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 583, "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/dev/parabola/problem_0583.png", "answer": "4+2*sqrt(2)", "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};Abs(LineSegmentOf(A, B)) = 8;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 584, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0584.png", "answer": "sqrt(10)/3\nx+pm*3*y=0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2 = 1);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 585, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 586, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0586.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/6 + x^2/m = 1);m: Number;FocalLength(G) = 6", "coords": {} }, { "index": 587, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 5.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0587.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 100)", "coords": {} }, { "index": 588, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0588.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;G: Circle;Expression(G) = (y^2 + (x - 2)^2 = 2);IsIntersect(Asymptote(C), G);L1: Line;L2: Line;Asymptote(C) = {L1, L2};Length(InterceptChord(L1, G)) = 2;Length(InterceptChord(L2, G)) = 2", "coords": {} }, { "index": 589, "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/dev/ellipse/problem_0589.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 590, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0590.png", "answer": "1/8", "fact_expr": "P: Point;PointOnCurve(P,RightPart(G)) = True;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;Center(H) = O;O: Origin;Radius(H) = a;TangentPoint(LineOf(P,F1),H) = A;A: Point;PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(P,F1))) = True;IsTangent(LineOf(P,F1),H) = True", "coords": {} }, { "index": 591, "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/dev/parabola/problem_0591.png", "answer": "(-oo,2]", "fact_expr": "G: Parabola;a: Real;Expression(G) = (x^2 = 4*y);A:Point;Coordinate(A)=(0,a);WhenMin(Distance(Vertex(G),A))", "coords": {} }, { "index": 592, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0592.png", "answer": "0", "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;b>0;Expression(G) = (x^2/2 - y^2/b^2 = 1);y0:Number;Coordinate(P) = (sqrt(3), y0);LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G))) = (y = x);PointOnCurve(P, G)", "coords": {} }, { "index": 593, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 594, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0594.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;p: Number;F: Point;M: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(C) = F;Intersection(Directrix(C),axis)=A;PointOnCurve(M, C);WhenMin(Abs(LineSegmentOf(M,F))/Abs(LineSegmentOf(M,A)))", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 595, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0595.png", "answer": "(x^2/16+y^2/12=1)&Negation(y=0)", "fact_expr": "O: Circle;Expression(O) = (x^2 + y^2 = 16);A: Point;Coordinate(A) = (-2, 0);B: Point;Coordinate(B) = (2, 0);IsTangent(l, O) = True;l: Line;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;G: Parabola;Directrix(G) = l", "coords": {} }, { "index": 596, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0596.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, G);F: Point;Focus(G) = F;O: Origin;Area(TriangleOf(O, A, F)) = p;Abs(LineSegmentOf(A, F)) = 4", "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/dev/ellipse/problem_0597.png", "answer": "[1,5)", "fact_expr": "G: Ellipse;t: Number;H: Line;k: Real;Expression(G) = (x^2/5 + y^2/t = 1);Expression(H) = (y = k*x + 1);PointOnCurve(Focus(G),xAxis);IsIntersect(G,H)", "coords": {} }, { "index": 598, "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/dev/ellipse/problem_0598.png", "answer": "4", "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/b^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = sqrt(3)/2;PointOnCurve(P,C) = True;P: Point;AngleOf(F1, P, F2) = ApplyUnit(90, degree);b:Number;4>b;b>0", "coords": {} }, { "index": 599, "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/dev/circle/problem_0599.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Circle;B: Point;A: Point;Endpoint(LineSegmentOf(A,B))={A,B};Expression(G) = (x^2 + y^2 = 1);Coordinate(B)=(4,0);PointOnCurve(A, G)", "coords": {} }, { "index": 600, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.5, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0600.png", "answer": "sqrt(3)/2", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (4*x^2 - 2*y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 601, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0601.png", "answer": "(1, 2)", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + (y - 2)^2 = 1);NumIntersection(Asymptote(G), H) = 0", "coords": {} }, { "index": 602, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 603, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0603.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z: Line;PointOnCurve(OneOf(Focus(G)), Z);IsPerpendicular(Z, OneOf(Asymptote(G)));H: Curve;Expression(H) = (x^2/b^2 + y^2/a^2 = 1);PointOnCurve(FootPoint(Z, OneOf(Asymptote(G))), H)", "coords": {} }, { "index": 604, "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/dev/ellipse/problem_0604.png", "answer": "(1/2, 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;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Negation(P=Endpoint(MajorAxis(C)));M: Point;PointOnCurve(M, LineSegmentOf(P, F1));O: Origin;VectorOf(M, F1) = 2*VectorOf(P, M);DotProduct(VectorOf(M, F2), VectorOf(O, P)) = 0", "coords": {} }, { "index": 605, "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/dev/parabola/problem_0605.png", "answer": "16/3", "fact_expr": "m: Line;E: Parabola;A: Point;F: Point;P: Point;Q: Point;B: Point;l: Line;Expression(E) = (y^2 = 4*x);Focus(E) = F;Directrix(E) = l;PointOnCurve(F, m);Intersection(m, E) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A,F)),l)=P;Intersection(PerpendicularBisector(LineSegmentOf(A,F)),xAxis)=Q;AngleOf(A,F,P)=AngleOf(A,F,Q)", "coords": {} }, { "index": 606, "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/dev/ellipse/problem_0606.png", "answer": "a^2/b^2", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Circle;B: Point;A: Point;N: Point;M: Point;P:Point;a > b;b > 0;L1:Line;L2:Line;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(O)=(x^2+y^2=b^2);PointOnCurve(P,G);Negation(P=Vertex(G));TangentOfPoint(P,O)={L1,L2};TangentPoint(L1,O)=A;TangentPoint(L2,O)=B;Intersection(LineOf(A,B), xAxis) = M;Intersection(LineOf(A,B), yAxis) = N", "coords": {} }, { "index": 607, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0607.png", "answer": "4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;MidPoint(LineSegmentOf(F, N)) = M;Abs(LineSegmentOf(F, N)) = 6", "coords": {} }, { "index": 608, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 609, "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/dev/ellipse/problem_0609.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(H)=Focus(G);Focus(H)=Vertex(G)", "coords": {} }, { "index": 610, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 611, "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/dev/ellipse/problem_0611.png", "answer": "3/5", "fact_expr": "G: Ellipse;a: Number;b: Number;m: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (m, 4);PointOnCurve(P, G);Focus(G) = {F1, F2};Radius(InscribedCircle(TriangleOf(P, F1, F2))) = 3/2", "coords": {} }, { "index": 612, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.000784158706665, "b": 1.6304504558913928, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 6.201316546158653e-06, "converged": true, "iterations": 81 }, "output_path": "results/dev/ellipse/problem_0612.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Coordinate(P) = (1, 3/2);Focus(C) = {F1, F2};PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4", "coords": { "P": [ 1.0, 1.5 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 613, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 614, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0614.png", "answer": "7", "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);Expression(Asymptote(G))=(4*x+pm*3*y=0);PointOnCurve(P,G);LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(P, F1, F2)) = 16;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 615, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 616, "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/dev/ellipse/problem_0616.png", "answer": "sqrt(2)/2", "fact_expr": "Slope(l) = sqrt(2)/2;l: Line;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(l, G) = {A, B};A: Point;B: Point;Focus(G) = {Projection(A, xAxis), Projection(B, xAxis)}", "coords": {} }, { "index": 617, "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/dev/hyperbola/problem_0617.png", "answer": "x^2/8 - y^2/18 = 1", "fact_expr": "G: Hyperbola;H: Hyperbola;M: Point;Expression(G) = (x^2/4 - y^2/9 = 1);Coordinate(M) = (2*sqrt(6), 6);PointOnCurve(M,H);Asymptote(G) = Asymptote(H)", "coords": { "M": [ 4.898979485566356, 6.0 ] } }, { "index": 618, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0618.png", "answer": "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;M: Point;Coordinate(M) = (1, 8);PointOnCurve(M, G);G: Line;Intersection(G, E) = {A, B};B: Point;A: Point;MidPoint(LineSegmentOf(A, B)) = M;IsPerpendicular(LineOf(A, B), OneOf(Asymptote(E))) = True", "coords": { "M": [ 1.0, 8.0 ] } }, { "index": 619, "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/dev/parabola/problem_0619.png", "answer": "4", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (3, y1);PointOnCurve(P,G);y1:Number", "coords": {} }, { "index": 620, "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/dev/parabola/problem_0620.png", "answer": "65/16", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/8);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;YCoordinate(M) = 2", "coords": {} }, { "index": 621, "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/dev/parabola/problem_0621.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l,G) = {A,B};Distance(MidPoint(LineSegmentOf(A,B)), yAxis) = 2", "coords": {} }, { "index": 622, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0622.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;P: Point;PointOnCurve(P, G) ;L: Line;PointOnCurve(P,L);IsParallel(L,RealAxis(G)) ;L1: Line;L2: Line;Asymptote(G) = {L1,L2};Intersection(L,L1) = M;M: Point;Intersection(L,L2) = N;N: Point;DotProduct(VectorOf(P, M), VectorOf(P, N)) = 2*b^2", "coords": {} }, { "index": 623, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 624, "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/dev/ellipse/problem_0624.png", "answer": "(0,sqrt(2)]", "fact_expr": "E: Ellipse;A: Point;B: Point;F: Point;P: Point;t:Number;Expression(E) = (x^2/2 + y^2 = 1);Coordinate(P)=(2,t);LeftFocus(E) = F;TangentOfPoint(P,E)={LineOf(P,A),LineOf(P,B)};TangentPoint(LineOf(P,B),E)=A;TangentPoint(LineOf(P,A),E)=B", "coords": {} }, { "index": 625, "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/dev/hyperbola/problem_0625.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 626, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0626.png", "answer": "sqrt(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;l: Line;PointOnCurve(F1, l);Inclination(l) = pi/6;A: Point;Intersection(l, LeftPart(G)) = A;B: Point;Intersection(l, RightPart(G)) = B;Abs(LineSegmentOf(A, F2)) = Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 627, "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/dev/ellipse/problem_0627.png", "answer": "9", "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, 0);Coordinate(B) = (2, 0);PointOnCurve(P, G);Abs(LineSegmentOf(P, A)) - Abs(LineSegmentOf(P, B)) = 2", "coords": { "A": [ -2.0, 0.0 ], "B": [ 2.0, 0.0 ] } }, { "index": 628, "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/dev/ellipse/problem_0628.png", "answer": "x^2/2+y^2=1", "fact_expr": "C: Ellipse;G:Line;a: Number;b: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x - y + 2*sqrt(2) = 0);Eccentricity(C) = sqrt(2)/2;Distance(RightVertex(C),G)=3", "coords": {} }, { "index": 629, "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/dev/ellipse/problem_0629.png", "answer": "8;4;(0,pm*2*sqrt(3));{(pm*2,0),(0,pm*4)};sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/16 = 1)", "coords": {} }, { "index": 630, "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/dev/ellipse/problem_0630.png", "answer": "-1/2", "fact_expr": "P: Point;Coordinate(P) = (1, 1);PointOnCurve(P, l) = True;l: Line;G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 631, "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/dev/parabola/problem_0631.png", "answer": "12", "fact_expr": "C: Parabola;P: Point;G: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C);Coordinate(G) = (6, 12)", "coords": { "G": [ 6.0, 12.0 ] } }, { "index": 632, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0632.png", "answer": "x^2/20 - y^2/16 = 1", "fact_expr": "C: Hyperbola;a: Number;a > 0;Expression(C) = (x^2/a^2 - y^2/16 = 1);Distance(RightVertex(C), OneOf(Asymptote(C))) = 4*sqrt(5)/3", "coords": {} }, { "index": 633, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 634, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0634.png", "answer": "1/8", "fact_expr": "G: Parabola;m: Number;H: Ellipse;Expression(G) = (y = m*x^2);Expression(H) = (x^2/2 + y^2/6 = 1);Focus(G) = UpperFocus(H)", "coords": {} }, { "index": 635, "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/dev/parabola/problem_0635.png", "answer": "9/8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 3*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin", "coords": {} }, { "index": 636, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0636.png", "answer": "x^2/16 + y^2/15 = 1", "fact_expr": "M:Circle;C1:Circle;Expression(C1) = ((x+1)^2 + y^2 = 36);IsInTangent(M,C1);C2: Circle;Expression(C2) = ((x-1)^2 + y^2 = 4);IsOutTangent(M,C2);Center(M) = M1;M1: Point", "coords": {} }, { "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/dev/parabola/problem_0637.png", "answer": "10", "fact_expr": "A: Point;Coordinate(A) = (8, 2);G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True", "coords": { "A": [ 8.0, 2.0 ] } }, { "index": 638, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 639, "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/dev/ellipse/problem_0639.png", "answer": "x^2/6+y^2/2=1", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F1: Point;F2: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(O,l);Inclination(l)=ApplyUnit(30,degree);OneOf(Intersection(l,C))=A;IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2));Area(TriangleOf(F1,A,F2)) = 2", "coords": {} }, { "index": 640, "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/dev/hyperbola/problem_0640.png", "answer": "sqrt(2)/2", "fact_expr": "C2: Hyperbola;C1: Ellipse;a:Number;b:Number;a>b;b>0;P: Point;F2: Point;Expression(C1) = (x^2/a^2+y^2/b^2=1);Expression(C2) = (x^2-y^2=4);RightFocus(C1)=F2;RightFocus(C2)=F2;Intersection(C1, C2)=P;Quadrant(P) = 1;Abs(LineSegmentOf(P, F2)) = 2", "coords": {} }, { "index": 641, "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/dev/hyperbola/problem_0641.png", "answer": "5/4", "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G), xAxis);Expression(Asymptote(G)) = (y = pm*(3/4)*x)", "coords": {} }, { "index": 642, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0642.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;H: Point;Coordinate(H) = (1, 2);PointOnCurve(H, Asymptote(G))", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 643, "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/dev/parabola/problem_0643.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 644, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0644.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": 645, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/dev/hyperbola/problem_0645.png", "answer": "x^2/9 - y^2 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1, F2};Expression(Asymptote(G)) = (y = pm*(x/3))", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 646, "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/dev/hyperbola/problem_0646.png", "answer": "2", "fact_expr": "G: Hyperbola;O: Origin;P: Point;F: Point;Expression(G) = (x^2 - y^2/3 = 1);RightFocus(G) = F;PointOnCurve(P,Asymptote(G));Abs(LineSegmentOf(O, P)) = 2", "coords": {} }, { "index": 647, "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/dev/hyperbola/problem_0647.png", "answer": "2", "fact_expr": "G: Hyperbola;H: Ellipse;a: Number;Expression(G) = (x^2/2 - y^2 = 1);a>0;Expression(H) = (y^2 + x^2/a^2 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 648, "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/dev/ellipse/problem_0648.png", "answer": "3", "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(A) = (1, 1);LeftFocus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 649, "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/dev/hyperbola/problem_0649.png", "answer": "10", "fact_expr": "C2: Hyperbola;C1: Ellipse;A: Point;Expression(C1) = (x^2/9 + y^2/4 = 1);Expression(C2) = (x^2/4 - y^2 = 1);OneOf(Intersection(C1, C2)) = A;F1: Point;F2: Point;Focus(C1) = {F1, F2};Distance(A, F1) + Distance(A, F2) = m ;m: Number;n: Number;Abs(Distance(A, F1) - Distance(A, F2)) = n", "coords": {} }, { "index": 650, "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/dev/circle/problem_0650.png", "answer": "2", "fact_expr": "l: Line;Expression(l) = (y = k*(x + 4));k:Number;G: Circle;Expression(G) = (y^2 + (x + 2)^2 = 4);A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;H: Line;Expression(H) = (3*x + 4*y - 6 = 0)", "coords": {} }, { "index": 651, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0651.png", "answer": "sqrt(2)", "fact_expr": "G:Hyperbola;b: Number;a: Number;F: Point;O: Origin;A: Point;c:Number;L:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;Coordinate(F) = (c, 0);PointOnCurve(F,L);IsPerpendicular(OneOf(Asymptote(G)),L);FootPoint(OneOf(Asymptote(G)),L)=A;Area(TriangleOf(O, A, F)) = c^2/4", "coords": {} }, { "index": 652, "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/dev/parabola/problem_0652.png", "answer": "(-2,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)", "coords": {} }, { "index": 653, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0653.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;L: Line;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), L);IsPerpendicular(L,xAxis);Intersection(L,G) = {N, M};IsDiameter(LineSegmentOf(M,N),H);PointOnCurve(LeftVertex(G),H)", "coords": {} }, { "index": 654, "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/dev/ellipse/problem_0654.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);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(P, F2))", "coords": {} }, { "index": 655, "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/dev/parabola/problem_0655.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = -4*x);F: Point;Focus(C) = F;A: Point;Coordinate(A) = (-2, 1);P: Point;PointOnCurve(P, C)", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 656, "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/dev/parabola/problem_0656.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;b: Number;a: Number;F: Point;Expression(G) = (y^2 = 4*x);a>0;b>0;Expression(H) = (b + a*x + y - 1 = 0);Focus(G) = F;PointOnCurve(F, H)", "coords": {} }, { "index": 657, "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/dev/parabola/problem_0657.png", "answer": "3", "fact_expr": "C: Parabola;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(M, C);XCoordinate(M) = 2", "coords": {} }, { "index": 658, "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/dev/parabola/problem_0658.png", "answer": "{9/16, 3/2}", "fact_expr": "C: Parabola;A: Point;M: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (2, 3/2);Focus(C) = F;PointOnCurve(M, C);WhenMin(Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,F)))", "coords": { "A": [ 2.0, 1.5 ] } }, { "index": 659, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0659.png", "answer": "English:", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))", "coords": {} }, { "index": 660, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0660.png", "answer": "pm*(3/4)", "fact_expr": "l: Line;C: Parabola;p: Number;k:Number;A: Point;B: Point;M: Point;F:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(M, l);Intersection(Directrix(C), xAxis) = M;Slope(l)=k;Intersection(l,C)={A,B};Abs(LineSegmentOf(A, M)) = (5/4)*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 661, "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/dev/ellipse/problem_0661.png", "answer": "x-2*y+3=0", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (-1,1)", "coords": {} }, { "index": 662, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0662.png", "answer": "y = pm*(sqrt(5)/2)*x", "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 663, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0663.png", "answer": "2*sqrt(2)", "fact_expr": "G1: Hyperbola;Expression(G1) = (-y^2/b^2 + x^2/a^2 = 1);G2: Hyperbola;Expression(G2) = (y^2/b^2 - x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;e1: Number;e2: Number;Eccentricity(G1) = e1;Eccentricity(G2) = e2", "coords": {} }, { "index": 664, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 665, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 666, "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/dev/hyperbola/problem_0666.png", "answer": "x^2/2 - y^2/4 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;B: Point;A: Point;D: Point;F2: Point;L:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 2);PointOnCurve(P,C);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F2,L);IsPerpendicular(L,xAxis);Intersection(L,C)={A,B};Intersection(LineSegmentOf(F1, B), yAxis) = D;IsPerpendicular(LineSegmentOf(A, D), LineSegmentOf(F1, B))", "coords": { "P": [ 2.0, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 667, "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/dev/ellipse/problem_0667.png", "answer": "sqrt(3)/3", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;B: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (1, 2);Slope(l)=-1/3;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 668, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0668.png", "answer": "x^2/9 - y^2/36 = 1", "fact_expr": "G: Hyperbola;a: Number;b: Number;Q: Point;F: Point;M: Point;N: Point;l:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;PointOnCurve(F,l);IsPerpendicular(l,OneOf(Asymptote(G)));FootPoint(l,OneOf(Asymptote(G)))=Q;Intersection(LineOf(F,Q), LeftPart(G)) = M;Intersection(LineOf(F,Q), RightPart(G)) = N;Abs(LineSegmentOf(M, Q)) = 3*Abs(LineSegmentOf(Q, N));Abs(LineSegmentOf(F, N)) = 4", "coords": {} }, { "index": 669, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/circle/problem_0669.png", "answer": "2*x-3*y+25=0", "fact_expr": "C: Circle;Expression(C) = (x^2 + y^2 = 25);M: Point;l: Line;Coordinate(M) = (-2, 3);PointOnCurve(M, l);A: Point;B: Point;Intersection(l, C) = {A, B};L1: Line;L2: Line;TangentOnPoint(A,C)=L1;TangentOnPoint(B,C)=L2;N: Point;Intersection(L1, L2) = N", "coords": {} }, { "index": 670, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 671, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 672, "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/dev/parabola/problem_0672.png", "answer": "(1, 1/4)", "fact_expr": "G: Parabola;Q: Point;P: Point;F: Point;Expression(G) = (y = x^2/4);Coordinate(Q) = (1, 1);Focus(G) = F;PointOnCurve(P, G);WhenMin(Perimeter(TriangleOf(P, Q, F)))", "coords": { "Q": [ 1.0, 1.0 ] } }, { "index": 673, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0673.png", "answer": "x^2-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;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = b;FocalLength(G) = 2*sqrt(5)", "coords": {} }, { "index": 674, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0674.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;l: Line;A: Point;B: Point;F: Point;M: Point;N: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/3;Directrix(G) = l;MidPoint(LineSegmentOf(A, B)) = M;Projection(M, l) = N", "coords": {} }, { "index": 675, "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/dev/hyperbola/problem_0675.png", "answer": "pm*sqrt(35)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);k: Number;Slope(G) = k;A: Point;B: Point;Intersection(G, RightPart(C)) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 676, "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/dev/ellipse/problem_0676.png", "answer": "3/5", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 1", "coords": {} }, { "index": 677, "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/dev/hyperbola/problem_0677.png", "answer": "[4 + sqrt(61), +oo)", "fact_expr": "G: Hyperbola;H: Circle;M: Point;F1: Point;F2: Point;A: Point;Expression(H) = (x^2 + (y - 6)^2 = 4);Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, H);PointOnCurve(M, RightPart(G))", "coords": {} }, { "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/dev/hyperbola/problem_0678.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Coordinate(LeftFocus(G))=(-2,0)", "coords": {} }, { "index": 679, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 680, "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/dev/parabola/problem_0680.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);P: Point;Coordinate(P) = (3/2, 1);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.5, 1.0 ] } }, { "index": 681, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 682, "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/dev/parabola/problem_0682.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;H: Line;Expression(H) = (x + 2 = 0);Distance(P, H) = 5;F: Point;Focus(G) = F", "coords": {} }, { "index": 683, "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/dev/parabola/problem_0683.png", "answer": "(0, -1)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -4*y)", "coords": {} }, { "index": 684, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 685, "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/dev/ellipse/problem_0685.png", "answer": "sqrt(21)/9", "fact_expr": "A: Point;B: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;VectorOf(A, F1) = 2*VectorOf(F1, B);AngleOf(F1, A, F2) = pi/3", "coords": {} }, { "index": 686, "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/dev/parabola/problem_0686.png", "answer": "12", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 3*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 687, "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/dev/parabola/problem_0687.png", "answer": "{x+y-1=0,x-y-1=0}", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 688, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0688.png", "answer": "5/3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};A: Point;x0: Number;Coordinate(A) = (x0, 4);x0>0;PointOnCurve(A, C);Area(TriangleOf(F1, A, F2)) = 20;e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 689, "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/dev/parabola/problem_0689.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);H: Circle;Expression(H) = (x^2 + (y - 3)^2 = 1);L: Line;TangentOfPoint(P, H) = L;A: Point;TangentPoint(L, H) = A", "coords": {} }, { "index": 690, "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/dev/ellipse/problem_0690.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;G: Line;A: Point;F: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(G) = (x = 2);RightFocus(C)=F;PointOnCurve(A, G);Intersection(LineSegmentOf(A,F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 691, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0691.png", "answer": "(2/3)*sqrt(6)", "fact_expr": "G: Hyperbola;a: Number;b: Number;e:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(G)))=pi/3;Eccentricity(G)=e", "coords": {} }, { "index": 692, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "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/dev/ellipse/problem_0692.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Ellipse;C:Curve;H: Point;Coordinate(H) = (-3, 2);PointOnCurve(H, G);Expression(C)=(x^2/9 + y^2/4 = 1);Focus(G)=Focus(C)", "coords": { "H": [ -3.0, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 693, "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/dev/ellipse/problem_0693.png", "answer": "(2, 4)", "fact_expr": "G: Ellipse;b: Real;H: Line;k: Number;b>0;Expression(G) = (x^2/16 + y^2/b^2 = 1);Expression(H) = (y = k*x + 2);PointOnCurve(Focus(G), xAxis);NumIntersection(H, G) = 2", "coords": {} }, { "index": 694, "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/dev/parabola/problem_0694.png", "answer": "-4", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), l);Intersection(l, G) = {A,B};x1:Number;y1:Number;x2:Number;y2:Number", "coords": {} }, { "index": 695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0695.png", "answer": "(pm*sqrt(7),0)", "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": 696, "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/dev/parabola/problem_0696.png", "answer": "y=-1/8", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 697, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0697.png", "answer": "7/5", "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, RightPart(G));Abs(LineSegmentOf(P, F1)) = 6*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e", "coords": {} }, { "index": 698, "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/dev/ellipse/problem_0698.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;D: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Intersection(RightDirectrix(G), xAxis) = D;PointOnCurve(P,G);AngleOf(P,F,D)=ApplyUnit(60,degree);Sin(AngleOf(P, D, F)) = 3/5", "coords": {} }, { "index": 699, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 700, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 701, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 702, "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/dev/hyperbola/problem_0702.png", "answer": "21", "fact_expr": "E: Ellipse;Expression(E) = (x^2/25 + y^2/16 = 1);G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};Focus(E) = {F1, F2};P: Point;OneOf(Intersection(E, G)) = P", "coords": {} }, { "index": 703, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 704, "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/dev/ellipse/problem_0704.png", "answer": "8", "fact_expr": "l: Line;G: Ellipse;F1: Point;A: Point;B: Point;Expression(G) = (x^2/2 + y^2/4 = 1);OneOf(Focus(G)) = F1;Negation(PointOnCurve(F1, l));Intersection(l, G) = {A, B}", "coords": {} }, { "index": 705, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 706, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 707, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0707.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);Expression(Asymptote(G))=(y=pm*x);Coordinate(OneOf(Focus(G))) = (4*sqrt(2), 0)", "coords": {} }, { "index": 708, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0708.png", "answer": "m - a^2", "fact_expr": "H: Ellipse;Expression(H) = (y^2/n + x^2/m = 1);m: Number;n: Number;m > n;n > 0;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(H) = F1;RightFocus(H) = F2;LeftFocus(G) = F1;RightFocus(G)= F2;P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 709, "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/dev/ellipse/problem_0709.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;F: Point;LeftFocus(G) = F;Coordinate(F) = (-c, 0);c: Number;H: Circle;Center(H) = F;Radius(H) = c;NumIntersection(H,LeftDirectrix(G)) = 2", "coords": {} }, { "index": 710, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 711, "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/dev/ellipse/problem_0711.png", "answer": "[-11/5,1]", "fact_expr": "A: Point;B: Point;G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};LeftVertex(G)=A;UpperVertex(G)=B;PointOnCurve(P, LineSegmentOf(A, B))", "coords": {} }, { "index": 712, "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/dev/hyperbola/problem_0712.png", "answer": "sqrt(13)/2, sqrt(13)/3", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/2)*x)", "coords": {} }, { "index": 713, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0713.png", "answer": "y=pm*(sqrt(3)/3*x)", "fact_expr": "G: Hyperbola;C:Parabola;b: Number;Expression(C)=(y^2 = 16*x);Expression(G) = (x^2/12 - y^2/b^2 = 1);Focus(C) = RightFocus(G)", "coords": {} }, { "index": 714, "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/dev/parabola/problem_0714.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;a: Number;G: Parabola;l: Line;e: Number;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 = 8*x);Directrix(G) = l;IsTangent(l, C);Eccentricity(C) = e", "coords": {} }, { "index": 715, "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/dev/hyperbola/problem_0715.png", "answer": "y^2/2 - x^2/4 = 1", "fact_expr": "G: Hyperbola;M: Point;C:Hyperbola;Expression(G) = (x^2 - 2*y^2 = 2);Coordinate(M) = (2,-2);Asymptote(C) = Asymptote(G);PointOnCurve(M,C)", "coords": { "M": [ 2.0, -2.0 ] } }, { "index": 716, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 717, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 718, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0718.png", "answer": "y^2=8*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;G: Circle;PointOnCurve(Center(G), C);O: Origin;PointOnCurve(O, G);IsTangent(G, Directrix(C));H: Line;Expression(H) = (x - sqrt(2)*y = 0);Length(InterceptChord(H, G)) = 2*sqrt(6)", "coords": {} }, { "index": 719, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0719.png", "answer": "sqrt(5)", "fact_expr": "C:Hyperbola;l:Line;a,b:Number;e:Number;a>0;b>0;Expression(C)=(x^2/a^2-y^2/b^2=1);Expression(l)=(x-2*y+2020=0);IsPerpendicular(OneOf(Asymptote(C)),l)=True;Eccentricity(C)=e", "coords": {} }, { "index": 720, "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/dev/ellipse/problem_0720.png", "answer": "sqrt(3)-1", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;LeftFocus(G) = F1;RightFocus(G) = F2;OneOf(Intersection(H, G)) = P;P: Point;IsDiameter(LineSegmentOf(F1, F2), H);H: Circle;AngleOf(P, F1, F2) = 2*AngleOf(P, F2, F1)", "coords": {} }, { "index": 721, "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/dev/parabola/problem_0721.png", "answer": "3*sqrt(2)", "fact_expr": "G: Parabola;H: Circle;F: Point;P: Point;Expression(G) = (y = x^2/8);Expression(H) = ((x + 3)^2 + (y + 3)^2 = 16);Focus(G) = F;L: Line;TangentOfPoint(F, H) = L;TangentPoint(H, L) = P", "coords": {} }, { "index": 722, "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/dev/ellipse/problem_0722.png", "answer": "4\n1", "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2}", "coords": {} }, { "index": 723, "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/dev/ellipse/problem_0723.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);x1: Number;y1: Number;P: Point;Coordinate(P) = (x1, y1);PointOnCurve(P, G);A: Point;Coordinate(A) = (3, 0);M: Point;Abs(VectorOf(A, M)) = 1;DotProduct(VectorOf(P, M), VectorOf(A, M)) = 0", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 724, "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/dev/ellipse/problem_0724.png", "answer": "x^2/36 - y^2/12 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/64 + y^2/16 = 1);Focus(G)=Focus(H);Expression(Asymptote(G))=(x+pm*sqrt(3)*y=0)", "coords": {} }, { "index": 725, "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/dev/ellipse/problem_0725.png", "answer": "2*sqrt(5)", "fact_expr": "C: Ellipse;a: Number;P: Point;F1:Point;F2:Point;x0:Number;a>2;Expression(C) = (y^2/4 + x^2/a^2 = 1);Coordinate(P) = (x0, 1);PointOnCurve(P, C);Focus(C)={F1,F2};Area(TriangleOf(P,F1,F2))=1", "coords": {} }, { "index": 726, "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/dev/ellipse/problem_0726.png", "answer": "2", "fact_expr": "C: Ellipse;a: Number;b: Real;F1: Point;P: 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);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(F1, P, F2)) = 4", "coords": {} }, { "index": 727, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 728, "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/dev/ellipse/problem_0728.png", "answer": "3/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)", "coords": {} }, { "index": 729, "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/dev/ellipse/problem_0729.png", "answer": "sqrt(5);3", "fact_expr": "G: Ellipse;O: Origin;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(M, G)", "coords": {} }, { "index": 730, "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/dev/parabola/problem_0730.png", "answer": "(1/2, +oo)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Negation(IsPerpendicular(LineSegmentOf(A, B), xAxis))", "coords": {} }, { "index": 731, "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/dev/ellipse/problem_0731.png", "answer": "7/3", "fact_expr": "G: Ellipse;H: Line;M: Point;O: Origin;B: Point;A: Point;Expression(G) = (x^2/9 + y^2/7 = 1);Slope(H)=-1/3;Negation(PointOnCurve(O,H));Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M", "coords": {} }, { "index": 732, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0732.png", "answer": "2", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;H: Point;I: Point;O: Origin;0 < a;a < b;c: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (a, 0);Coordinate(I) = (0, b);HalfFocalLength(G) = c;PointOnCurve(H, l);PointOnCurve(I, l);Distance(O, l) = (3/4)*c", "coords": {} }, { "index": 733, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 734, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0734.png", "answer": "sqrt(13)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;F2: Point;P: Point;M: Point;F1: 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);FootPoint(Asymptote(C),L)=M;Intersection(L,RightPart(C))=P;VectorOf(F2, P) = 2*VectorOf(P, M);AngleOf(F1, P, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 735, "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/dev/ellipse/problem_0735.png", "answer": "sqrt(2)-1", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;O: Origin;l: Line;PointOnCurve(O, l);Inclination(l) = ApplyUnit(60, degree);M: Point;OneOf(Intersection(l, C)) = M;Abs(VectorOf(M, F1) + VectorOf(M, F2)) = Abs(VectorOf(M, F1) - VectorOf(M, F2))", "coords": {} }, { "index": 736, "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/dev/hyperbola/problem_0736.png", "answer": "8", "fact_expr": "C1: Hyperbola;C2: Parabola;a:Number;b:Number;p:Number;p>0;a>0;b>0;Expression(C1)=(x^2/a^2-y^2/b^2=1);Eccentricity(C1)=2;Expression(C2)=(x^2=2*(p*y));Distance(Focus(C2),Asymptote(C1))=2", "coords": {} }, { "index": 737, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 738, "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/dev/ellipse/problem_0738.png", "answer": "{3,16/3}", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 739, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 740, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0740.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;y1: Number;y2: Number;Coordinate(A) = (1, y1);Coordinate(B) = (9, y2);PointOnCurve(A, G);PointOnCurve(B, G);y2>y1;y1>0;F: Point;Focus(G) = F;Abs(LineSegmentOf(B, F)) = 5*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 741, "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/dev/parabola/problem_0741.png", "answer": "(-oo,2]", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Q: Point;PointOnCurve(Q, G);P: Point;Coordinate(P) = (a, 0);a: Number;Abs(LineSegmentOf(P, Q)) >= Abs(a)", "coords": {} }, { "index": 742, "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/dev/ellipse/problem_0742.png", "answer": "2", "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Expression(G) = (x^2/2 + y^2 = 1);Center(G) = O;LeftFocus(G) = F;PointOnCurve(P, G)", "coords": {} }, { "index": 743, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0743.png", "answer": "8;x = -4", "fact_expr": "C: Parabola;p: Number;G: Line;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + 2*y - 4 = 0);PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 744, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 745, "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/dev/parabola/problem_0745.png", "answer": "pm*3*sqrt(2)/3", "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 3;Coordinate(N) = (-2, 0)", "coords": { "N": [ -2.0, 0.0 ] } }, { "index": 746, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 747, "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/dev/hyperbola/problem_0747.png", "answer": "y^2 = 4*sqrt(5)*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2 - y^2/4 = 1);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 748, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 749, "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/dev/ellipse/problem_0749.png", "answer": "(0, sqrt(6)/3]", "fact_expr": "F1: Point;F2: Point;b: Number;a: Number;a > b;b > 0;H: Line;P: Point;c: Number;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Expression(H) = (x = (2*a^2)/c);PointOnCurve(P, H);Abs(LineSegmentOf(P, F2)) = 2*c;LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 750, "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/dev/parabola/problem_0750.png", "answer": "2*sqrt(2)", "fact_expr": "F:Point;G:Parabola;Expression(G) = (x^2=4*y);B: Point;Coordinate(B) = (0,-1);A:Point;PointOnCurve(A,G) = True;WhenMin(Abs(LineSegmentOf(A,F)) / Abs(LineSegmentOf(A,B))) = True;Focus(G) = F", "coords": { "B": [ 0.0, -1.0 ] } }, { "index": 751, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0751.png", "answer": "2", "fact_expr": "G: Parabola;m: Real;Expression(G) = (y^2 = m*x);Coordinate(Focus(G))=(1/2,0)", "coords": {} }, { "index": 752, "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/dev/ellipse/problem_0752.png", "answer": "20", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (y^2/9 + x^2/a^2 = 1);Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;PointOnCurve(F1, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);a:Number", "coords": {} }, { "index": 753, "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/dev/ellipse/problem_0753.png", "answer": "x+2*y-8=0", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(MidPoint(InterceptChord(l,G)))=(4,2)", "coords": {} }, { "index": 754, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0754.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 755, "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/dev/hyperbola/problem_0755.png", "answer": "[-2/3,0)", "fact_expr": "F: Point;M: Hyperbola;Expression(M) = (x^2/9 - y^2/16 = 1);LeftFocus(M) = F;O: Origin;PointOnCurve(O, l) = True;l: Line;Intersection(l,LeftPart(M)) = A;Intersection(l, RightPart(M)) = B;A: Point;B: Point", "coords": {} }, { "index": 756, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0756.png", "answer": "sqrt(2)+1", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;D: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Intersection(Directrix(C), xAxis) = A;TangentOfPoint(A,C)=LineOf(A,B);TangentPoint(LineOf(A,B),C)=B;Quadrant(B)=1;PointOnCurve(D,C);Quadrant(D)=4;IsParallel(LineSegmentOf(D,F),LineSegmentOf(A,B))", "coords": {} }, { "index": 757, "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/dev/hyperbola/problem_0757.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 758, "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/dev/ellipse/problem_0758.png", "answer": "sqrt(3)", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 759, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 760, "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/dev/hyperbola/problem_0760.png", "answer": "x^2-y^2/4=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = 2*x);Distance(Focus(G), Asymptote(G)) = 2", "coords": {} }, { "index": 761, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0761.png", "answer": "{2,(2*sqrt(3)/3)}", "fact_expr": "F: Point;C: Hyperbola;a: Number;b: Number;a>0;b>0;l: Line;A: Point;B: Point;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(l, l1) = A;Intersection(l,l2) = B;Abs(VectorOf(A, F)) = 2*Abs(VectorOf(B, F))", "coords": {} }, { "index": 762, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0762.png", "answer": "4", "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (-y^2/3 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(3)/2))", "coords": {} }, { "index": 763, "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/dev/hyperbola/problem_0763.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/t^2 = 1);t: Real;t>0;H: Parabola;Expression(H) = (y = x^2/8);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 764, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 765, "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/dev/hyperbola/problem_0765.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(Center(H), G);PointOnCurve(RightVertex(G),H);PointOnCurve(RightFocus(G),H)", "coords": {} }, { "index": 766, "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/dev/parabola/problem_0766.png", "answer": "sqrt(5)", "fact_expr": "G: Line;Expression(G) = (y = (1/2)*x + 1);C: Parabola;Expression(C) = (x^2 = 4*y);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin", "coords": {} }, { "index": 767, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 768, "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/dev/parabola/problem_0768.png", "answer": "x - y + 2 = 0", "fact_expr": "G: Parabola;l: Line;P1: Point;P2: Point;Expression(G) = (x^2 = y);IsChordOf(LineSegmentOf(P1, P2), G);PerpendicularBisector(LineSegmentOf(P1, P2)) = l;Expression(l) = (y = x + 3)", "coords": {} }, { "index": 769, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 14.875267028808594, "direction": "right" }, "optimization": { "final_loss": 7.913284778594971, "converged": false, "iterations": 500 }, "output_path": "results/dev/parabola/problem_0769.png", "answer": "(-4,0)", "fact_expr": "G: Parabola;a: Number;H: Point;Expression(G) = (y^2 = 2*(a*x));Coordinate(H) = (-1, 4);PointOnCurve(H, G)", "coords": { "H": [ -1.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 770, "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/dev/hyperbola/problem_0770.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": 771, "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/dev/ellipse/problem_0771.png", "answer": "2", "fact_expr": "A1: Point;A2: Point;G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a>1;LeftVertex(G) = A1;RightVertex(G) = A2;P: Point;PointOnCurve(P, G) = True;k1: Number;Slope(LineOf(P,A1)) = k1;k2: Number;Slope(LineOf(P,A2)) = k2;k1 * k2 = -1/4;a: Real", "coords": {} }, { "index": 772, "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/dev/ellipse/problem_0772.png", "answer": "3", "fact_expr": "G: Ellipse;H: Line;m: Number;F1: Point;A: Point;B: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (x = m);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 773, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0773.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Line;A: Point;F1: Point;F2: Point;B: Point;a>b;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, G);Intersection(G,RightPart(C)) = {A, B};Abs(LineSegmentOf(A, F1)) = 2*Abs(LineSegmentOf(A, F2));AngleOf(A, F1, F2) = AngleOf(F1, B, F2)", "coords": {} }, { "index": 774, "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/dev/hyperbola/problem_0774.png", "answer": "1/2", "fact_expr": "P: Point;a: Number;b: Number;Coordinate(P) = (a, b);G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(P, RightPart(G));Distance(P,Asymptote(G)) = (sqrt(2))", "coords": {} }, { "index": 775, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 25.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0775.png", "answer": "x^2/12-y^2/8=1", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/5 = 1);Focus(H) = Focus(G);I: Point;Coordinate(I) = (3*sqrt(2), 2);PointOnCurve(I, G);G: Hyperbola", "coords": { "I": [ 4.242640687119286, 2.0 ] } }, { "index": 776, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0776.png", "answer": "y^2=2*x", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, l);Inclination(l) = pi/4;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 777, "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/dev/parabola/problem_0777.png", "answer": "{(4,4),(4,-4)}", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Focus(G)=F;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 778, "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/dev/ellipse/problem_0778.png", "answer": "3/4", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Intersection(l, G) = {A, B};B: Point;A: Point;M: Point;Coordinate(M)=(1,-1);MidPoint(LineSegmentOf(A,B)) = M", "coords": { "M": [ 1.0, -1.0 ] } }, { "index": 779, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0779.png", "answer": "4", "fact_expr": "G: Parabola;p: Real;H: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (-2, 4);PointOnCurve(H, Directrix(G))", "coords": { "H": [ -2.0, 4.0 ] } }, { "index": 780, "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/dev/ellipse/problem_0780.png", "answer": "(1/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (-c + x)^2 = c^2);c=sqrt(a^2-b^2);NumIntersection(H,G)=2", "coords": {} }, { "index": 781, "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/dev/parabola/problem_0781.png", "answer": "sqrt(3)", "fact_expr": "O: Origin;F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;Expression(l) = (y = m*(x - 1));m: Number;Intersection(l, G) = {A, B};A: Point;B: Point;Quadrant(A) = 1;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 782, "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/dev/hyperbola/problem_0782.png", "answer": "1", "fact_expr": "H: Parabola;Expression(H) = (x^2 = 8*y);G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 783, "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/dev/parabola/problem_0783.png", "answer": "5", "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (4, 2);Focus(C) = F;PointOnCurve(P, C)", "coords": { "A": [ 4.0, 2.0 ] } }, { "index": 784, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0784.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;O: Origin;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;L1:Line;L2:Line;Asymptote(G)={L1,L2};PointOnCurve(A,L1);PointOnCurve(B,L2);VectorOf(B,F)=2*VectorOf(F,A);DotProduct(VectorOf(O, A), VectorOf(A, B)) = 0", "coords": {} }, { "index": 785, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0785.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;M: Point;N: Point;C:Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G), H);IsPerpendicular(H,xAxis);Intersection(H, G) = {M, N};IsDiameter(LineSegmentOf(M,N),C);PointOnCurve(RightVertex(G),C)", "coords": {} }, { "index": 786, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 787, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/hyperbola/problem_0788.png", "answer": "sqrt(7)/2", "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) = (2, sqrt(3));PointOnCurve(H, OneOf(Asymptote(G)))", "coords": { "H": [ 2.0, 1.7320508075688772 ] } }, { "index": 789, "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/dev/ellipse/problem_0789.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 2", "coords": {} }, { "index": 790, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0790.png", "answer": "4/3", "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F1, l);l1, l2: Line;Asymptote(C) = {l1, l2};A: Point;B: Point;Intersection(l, l1) = A;Intersection(l, l2) = B;3*VectorOf(F1, A) = 2*VectorOf(A, B);DotProduct(VectorOf(F1, B), VectorOf(F2, B)) = 0", "coords": {} }, { "index": 791, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0791.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;A: Point;B: Point;PointOnCurve(A,G) = True;PointOnCurve(B, G) = True;AngleOf(A, F, B) = pi/2;IsChordOf(LineSegmentOf(A, B), G) = True;MidPoint(LineSegmentOf(A,B)) = M;M: Point;Projection(M,Directrix(G)) = N;N: Point", "coords": {} }, { "index": 792, "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/dev/ellipse/problem_0792.png", "answer": "((2-sqrt(2))/4, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;OneOf(Focus(G)) = F;H: Line;Expression(H) = (y = k*x);k: Number;A: Point;B: Point;Intersection(H, G) = {A, B};AngleOf(A, F, B) = ApplyUnit(135, degree);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 793, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 794, "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/dev/parabola/problem_0794.png", "answer": "16", "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) = sqrt(3)/3;Intersection(G,C)={A,B}", "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/dev/hyperbola/problem_0795.png", "answer": "sqrt(2)", "fact_expr": "E: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E)=F1;RightFocus(E)=F2;PointOnCurve(M, E);IsPerpendicular(LineSegmentOf(M,F1),xAxis);Sin(AngleOf(M, F2, F1)) = 1/3", "coords": {} }, { "index": 796, "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/dev/parabola/problem_0796.png", "answer": "1", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G) ;Focus(G) = F;F:Point", "coords": {} }, { "index": 797, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0797.png", "answer": "sqrt(7)", "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;M: Point;N: Point;PointOnCurve(M, LeftPart(C)) = True;PointOnCurve(N, RightPart(C)) = True;IsParallel(LineSegmentOf(M, N), LineSegmentOf(F1, F2)) = True;Abs(LineSegmentOf(M, N)) = Abs(LineSegmentOf(F1, F2))/2;Intersection(LineSegmentOf(F1, N), C) = Q;Q: Point;Abs(LineSegmentOf(F1, Q)) = (2/5)*Abs(LineSegmentOf(F1, N))", "coords": {} }, { "index": 798, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0798.png", "answer": "16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l) = True;l: Line;Intersection(l, LeftPart(G)) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 799, "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/dev/ellipse/problem_0799.png", "answer": "5/7", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/36 + y^2/b^2 = 1);b: Number;F1: Point;F2: Point;Focus(C1) = {F1, F2};C2: Parabola;Expression(C2) = (y^2 = 2*p*x);p: Number;p > 0;Focus(C2) = F2;P: Point;Intersection(C1, C2) = P;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 800, "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/dev/hyperbola/problem_0800.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)", "coords": {} }, { "index": 801, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 802, "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/dev/hyperbola/problem_0802.png", "answer": "ApplyUnit(90, degree)", "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);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32", "coords": {} }, { "index": 803, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 804, "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/dev/parabola/problem_0804.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 1);A: Point;B: Point;l: Line;PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "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/dev/ellipse/problem_0805.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(N) = (0, 2);Eccentricity(C) = sqrt(2)/2;IsIntersect(G,C) = True;Center(G) = N;Max(Radius(G)) = sqrt(26)", "coords": { "N": [ 0.0, 2.0 ] } }, { "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/dev/parabola/problem_0806.png", "answer": "(1, 1/4)", "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (x^2 = 4*y);Coordinate(Q) = (1, 2);PointOnCurve(P, G);WhenMin(Distance(P, Q) + Distance(P, Focus(G)))", "coords": { "Q": [ 1.0, 2.0 ] } }, { "index": 807, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0807.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;P: Point;O:Origin;Expression(G) = (y = x);Coordinate(P) = (2, 2);Vertex(C)=O;PointOnCurve(Focus(C), xAxis);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 808, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "vertical" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/dev/hyperbola/problem_0808.png", "answer": "y = pm*(sqrt(2)/4)*x", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/16 - x^2/m = 1);Eccentricity(G) = 3", "coords": {} }, { "index": 809, "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/dev/parabola/problem_0809.png", "answer": "4", "fact_expr": "C: Parabola;P: Point;F: Point;x0:Number;y0:Number;Expression(C) = (y^2 = 4*x);Coordinate(P) = (x0, y0);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 810, "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/dev/ellipse/problem_0810.png", "answer": "24/7", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(l) = (x - y + 1 = 0);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 811, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0811.png", "answer": "3/2", "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, C);Abs(LineSegmentOf(P, F)) = 5*a;AngleOf(P, F, O) = ApplyUnit(120, degree);O: Origin", "coords": {} }, { "index": 812, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0812.png", "answer": "x^2/2-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;Length(ImageinaryAxis(G)) = 2;FocalLength(G) = 2*sqrt(3)", "coords": {} }, { "index": 813, "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/dev/parabola/problem_0813.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (0, 1);PointOnCurve(P, G);Distance(P,Directrix(G))=d;d:Number", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 814, "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/dev/hyperbola/problem_0814.png", "answer": "2*sqrt(13)", "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Coordinate(H) = (-3*sqrt(2), 2);PointOnCurve(H, G)", "coords": { "H": [ -4.242640687119286, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 815, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0815.png", "answer": "4", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (-y^2/4 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = x)", "coords": {} }, { "index": 816, "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/dev/hyperbola/problem_0816.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 817, "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/dev/ellipse/problem_0817.png", "answer": "2*sqrt(3)", "fact_expr": "C: Curve;P: Point;Coordinate(P) = (x1, y1);Expression(C) = (x^2 + y^2/3 = 1);x1:Number;y1:Number;PointOnCurve(P, C)", "coords": {} }, { "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/dev/ellipse/problem_0818.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Ellipse;P: Point;M: Point;N: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Endpoint(MajorAxis(G))={M,N};PointOnCurve(P,G);Negation(M=P);Negation(N=P)", "coords": {} }, { "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/dev/parabola/problem_0819.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;A: Point;B: Point;M: Point;N: Point;F: Point;p>0;Focus(G)=F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;IsPerpendicular(LineSegmentOf(M,N),Directrix(G));FootPoint(LineSegmentOf(M,N),Directrix(G))=N;PointOnCurve(M, LineSegmentOf(M, N))", "coords": {} }, { "index": 820, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 821, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 822, "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/dev/ellipse/problem_0822.png", "answer": "sqrt(2)/2", "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;B: Point;O: Origin;M: Point;F1:Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(E)=A;UpperVertex(E)=B;LeftFocus(E)=F1;PointOnCurve(M, E);L:Line;PointOnCurve(M,L);IsPerpendicular(L,xAxis);PointOnCurve(F1,L);IsParallel(LineSegmentOf(A, B),LineSegmentOf(O,M))", "coords": {} }, { "index": 823, "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/dev/parabola/problem_0823.png", "answer": "2*y - 1 = 0", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*y)", "coords": {} }, { "index": 824, "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/dev/parabola/problem_0824.png", "answer": "2022", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2021, t);PointOnCurve(P, G);Focus(G) = F;t:Number", "coords": {} }, { "index": 825, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 826, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 827, "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/dev/ellipse/problem_0827.png", "answer": "(sqrt(2)-1, 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;c: Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);a/Sin(AngleOf(P, F1, F2)) = c/Sin(AngleOf(P, F2, F1))", "coords": {} }, { "index": 828, "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/dev/hyperbola/problem_0828.png", "answer": "(pm*sqrt(2), 0)\ny = pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 829, "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/dev/ellipse/problem_0829.png", "answer": "1/4", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);F1:Point;F2:Point;LeftFocus(G)=F1;RightFocus(G)=F2;a: Number;a>1;A:Point;B:Point;M:Point;N:Point;PointOnCurve(M, G);PointOnCurve(N, G);Max(Area(TriangleOf(M, A, B))) = 4*(1 + sqrt(2)) ;LeftVertex(G)=A;UpperVertex(G)=B", "coords": {} }, { "index": 830, "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/dev/hyperbola/problem_0830.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": 831, "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/dev/parabola/problem_0831.png", "answer": "sqrt(3)/2 + 1", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 832, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0832.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;p: Number;O: Origin;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));F: Point;Focus(G) = F;PointOnCurve(F, l);Slope(l) = 2;Intersection(l, G) = {A, B};Area(TriangleOf(O, A, B)) = sqrt(5)", "coords": {} }, { "index": 833, "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/dev/ellipse/problem_0833.png", "answer": "32", "fact_expr": "C: Ellipse;P: Point;F2: Point;F1: Point;Expression(C) = (x^2/25 + y^2/16 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);DotProduct(VectorOf(P, F1),VectorOf(P, F2))=0", "coords": {} }, { "index": 834, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 835, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 836, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 837, "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/dev/hyperbola/problem_0837.png", "answer": "8*sqrt(2)/3", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = x + 1)", "coords": {} }, { "index": 838, "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/dev/ellipse/problem_0838.png", "answer": "{2,25/2}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = e ;e:Number;e= sqrt(15)/5", "coords": {} }, { "index": 839, "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/dev/ellipse/problem_0839.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;H: Circle;P: Point;M: Point;x1:Number;y1:Number;l1:Line;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x - 3)^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);TangentOfPoint(P,H)=l1;TangentPoint(l1,H)=M", "coords": {} }, { "index": 840, "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/dev/hyperbola/problem_0840.png", "answer": "x^2/4 - y^2/5 = 1", "fact_expr": "C: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(C) = {F1,F2};G: Line;Expression(G) = (y = x*(sqrt(5)/2));OneOf(Asymptote(C)) = G", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 841, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0841.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;H: Line;N: Point;F: Point;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y = 2);Focus(G) = F;Intersection(H, yAxis) = M;Intersection(H,G)=N;4*Abs(LineSegmentOf(N, F)) = 5*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 842, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0842.png", "answer": "y^2 = -44*x", "fact_expr": "G: Parabola;a: Number;M: Point;a<0;Expression(G) = (y^2 = a*x);Coordinate(M) = (5, 3);Distance(M, Directrix(G)) = 6", "coords": { "M": [ 5.0, 3.0 ] } }, { "index": 843, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0843.png", "answer": "8", "fact_expr": "H: Parabola;Expression(H) = (x^2 = a*y);a: Number;G: Hyperbola;Expression(G) = (-x^2 + y^2 = 2);Focus(H) = UpperFocus(G)", "coords": {} }, { "index": 844, "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/dev/parabola/problem_0844.png", "answer": "(1/4, 1)", "fact_expr": "C: Parabola;B: Point;Q: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(B) = (4, 1);PointOnCurve(Q, C);Focus(C) = F;WhenMin(Distance(Q, B) + Distance(Q, F))", "coords": { "B": [ 4.0, 1.0 ] } }, { "index": 845, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9999431050898532, "b": 1.731428600448389, "orientation": "horizontal" }, "optimization": { "final_loss": 6.486633488920056e-05, "converged": true, "iterations": 80 }, "output_path": "results/dev/hyperbola/problem_0845.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "E: Hyperbola;Eccentricity(E) = 2;F1: Point;F2: Point;Focus(E) = {F1, F2};Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0)", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ] } }, { "index": 846, "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/dev/parabola/problem_0846.png", "answer": "16", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};PointOnCurve(Focus(C),l);Inclination(l)=pi/6", "coords": {} }, { "index": 847, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0847.png", "answer": "12", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;p: Real;Expression(G) = (x^2/(16 - m) - y^2/(m + 20) = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 848, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 849, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 850, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 851, "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/dev/hyperbola/problem_0851.png", "answer": "x^2/4 - y^2/16 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;l:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Asymptote(C))=l;Expression(l)=(y=2*x);Length(RealAxis(C))<6", "coords": {} }, { "index": 852, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 853, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0853.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Circle;Expression(H) = (y^2 + (x - 3)^2 = 16);IsTangent(Directrix(G), H)", "coords": {} }, { "index": 854, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 6.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/parabola/problem_0854.png", "answer": "y^2/9 - x^2/27 = 1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (x^2 = 24*y);OneOf(Focus(G)) = Focus(H);Inclination(OneOf(Asymptote(G)))=ApplyUnit(30,degree)", "coords": {} }, { "index": 855, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0855.png", "answer": "sqrt(5)/5", "fact_expr": "C: Parabola;p: Number;B: Point;A: Point;M: Point;F:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (2, -1);PointOnCurve(A, C);PointOnCurve(B, C);Focus(C)=F;Distance(A,F)+Distance(B,F)=6;MidPoint(LineSegmentOf(A,B)) = M;Negation(A=B)", "coords": { "M": [ 2.0, -1.0 ] } }, { "index": 856, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 857, "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/dev/hyperbola/problem_0857.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Inclination(OneOf(Asymptote(C)))=ApplyUnit(60,degree)", "coords": {} }, { "index": 858, "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/dev/hyperbola/problem_0858.png", "answer": "4/5", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(m + 4) + x^2/(2*m) = 1);m: Real;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "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/dev/hyperbola/problem_0859.png", "answer": "x^2/4-y^2/16=1\nsqrt(5)", "fact_expr": "C: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(C) = (-y^2/16 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 4", "coords": {} }, { "index": 860, "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/dev/ellipse/problem_0860.png", "answer": "9/4", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2);H: Ellipse;Expression(H) = (x^2/2 + y^2/m = 1);m: Number;Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 861, "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/dev/parabola/problem_0861.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, -1);H: Line;Expression(H) = (x = -1)", "coords": { "A": [ 0.0, -1.0 ] } }, { "index": 862, "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/dev/parabola/problem_0862.png", "answer": "2*x-y-1=0", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);Coordinate(P) = (1, 1);MidPoint(LineSegmentOf(A,B))=P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 863, "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/dev/parabola/problem_0863.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "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/dev/parabola/problem_0864.png", "answer": "(2, pm*2*sqrt(2))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 3", "coords": {} }, { "index": 865, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/ellipse/problem_0866.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;H: Line;Expression(H) = (-sqrt(a*b) + a*x + b*y = 0);G: Circle;Expression(G) = (x^2 + y^2 = 2/5);IsTangent(H, G)", "coords": {} }, { "index": 867, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 868, "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/dev/parabola/problem_0868.png", "answer": "1/2", "fact_expr": "G: Parabola;A: Point;F: Point;B: Point;lambda:Number;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", "coords": {} }, { "index": 869, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 870, "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/dev/parabola/problem_0870.png", "answer": "y= 3/2", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/6)", "coords": {} }, { "index": 871, "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/dev/hyperbola/problem_0871.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 4)", "coords": {} }, { "index": 872, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/parabola/problem_0872.png", "answer": "15/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 5*x);A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;Distance(A, Focus(G)) + Distance(B, Focus(G)) = 10", "coords": {} }, { "index": 873, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 874, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0874.png", "answer": "4", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};Negation(A=B);y0: Number;Coordinate(A) = (2, y0);Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 875, "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/dev/hyperbola/problem_0875.png", "answer": "x^2/4 - y^2/2 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);H: Point;Coordinate(H) = (2, 0);PointOnCurve(H, Z)", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 876, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0876.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": 877, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0877.png", "answer": "(pm*sqrt(7),0)", "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": 878, "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/dev/ellipse/problem_0878.png", "answer": "[2,6]", "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Center(G)=O;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G)=F;PointOnCurve(P, G)", "coords": {} }, { "index": 879, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 880, "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/dev/ellipse/problem_0880.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2/5 + y^2/p^2 = 1);PointOnCurve(LeftFocus(H),Directrix(G))", "coords": {} }, { "index": 881, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0881.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (x^2/9 - y^2/m^2 = 1);Distance(OneOf(Vertex(G)), OneOf(Asymptote(G))) = 1", "coords": {} }, { "index": 882, "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/dev/ellipse/problem_0882.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 883, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 884, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 885, "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/dev/ellipse/problem_0885.png", "answer": "(sqrt(5) - 1)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;P: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;LowerVertex(C) = B;PointOnCurve(F1, G);PointOnCurve(F2, G);PointOnCurve(B, G);Intersection(G, C) = P;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(F1, F2))", "coords": {} }, { "index": 886, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 887, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 888, "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/dev/hyperbola/problem_0888.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/2 + y^2/4 = 1)", "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/dev/hyperbola/problem_0889.png", "answer": "(1, 2)", "fact_expr": "G: Hyperbola;a: Number;b: 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);Negation(Vertex(G) = P);AngleOf(P, F2, F1) = 3*AngleOf(P, F1, F2)", "coords": {} }, { "index": 890, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0890.png", "answer": "x^2/3-y^2=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);Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(3)/3));PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 891, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 892, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 893, "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/dev/parabola/problem_0893.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);x1: Number;y1: Number;M: Point;Coordinate(M) = (x1, y1);PointOnCurve(M, G);Distance(M, Focus(G)) = 3;O: Origin", "coords": {} }, { "index": 894, "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/dev/parabola/problem_0894.png", "answer": "x^2/(1/16)-y^2/(1/4)=1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (y = x^2);Expression(OneOf(Asymptote(G)))=(x+2*y=0);NumIntersection(G,Directrix(H)) = 1", "coords": {} }, { "index": 895, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 896, "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/dev/hyperbola/problem_0896.png", "answer": "31", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = 15", "coords": {} }, { "index": 897, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0897.png", "answer": "(x - 4)^2 + (y - 4)^2 = 25", "fact_expr": "G: Parabola;p: Number;C: Circle;A: Point;F: Point;m: Number;p > 0;m > 0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (4, m);PointOnCurve(A, G);Focus(G) = F;Center(C) = A;Radius(C) = Abs(LineSegmentOf(A, F));Length(InterceptChord(C, yAxis)) = 6", "coords": {} }, { "index": 898, "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/dev/hyperbola/problem_0898.png", "answer": "2", "fact_expr": "G: Hyperbola;H: Circle;a: Number;Expression(G) = (x^2 - y^2 = 1);a > 0;Expression(H) = (y^2 + (x - 1)^2 = a^2);NumIntersection(G, H) = 3", "coords": {} }, { "index": 899, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0899.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Hyperbola;b: Number;a: Number;C: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (-8*x + x^2 + y^2 + 12 = 0);IsTangent(Asymptote(G), C);RightFocus(G) = Center(C)", "coords": {} }, { "index": 900, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0900.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 901, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0901.png", "answer": "sqrt(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;RightFocus(G) = F;O: Origin;PointOnCurve(O, l) = True;l: Line;Intersection(l, G) = {M, N};M: Point;N: Point;DotProduct(VectorOf(M, F), VectorOf(N, F)) = 0;Area(TriangleOf(M, N, F)) = a*b", "coords": {} }, { "index": 902, "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/dev/parabola/problem_0902.png", "answer": "4*sqrt(7)/7", "fact_expr": "P: Point;PointOnCurve(P,C1) = True;C1: Curve;Expression(C1) = (y^2 = 8*x);Q: Point;PointOnCurve(Q,C2) = True;C2: Curve;Expression(C2) = (y^2 + (x - 2)^2 = 1);O: Origin", "coords": {} }, { "index": 903, "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/dev/parabola/problem_0903.png", "answer": "3", "fact_expr": "K: Parabola;A: Point;B: Point;F: Point;Expression(K) = (x^2 = 4*y);PointOnCurve(A,K);PointOnCurve(B,K);Negation(YCoordinate(A)=YCoordinate(B));Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 4;Focus(K)=F", "coords": {} }, { "index": 904, "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/dev/hyperbola/problem_0904.png", "answer": "2", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))+Abs(LineSegmentOf(P,F2))=6", "coords": {} }, { "index": 905, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0905.png", "answer": "[sqrt(2), sqrt(3)+1]", "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;G: Line;Expression(G) = (y = k*x);k: Number;In(k, [sqrt(3)/3, sqrt(3)]);A: Point;B: Point;Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 906, "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/dev/hyperbola/problem_0906.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, RightPart(G));C1: Circle;Expression(C1) = ((x + 5)^2 + y^2 = 4);C2: Circle;Expression(C2) = ((x - 5)^2 + y^2 = r^2);r: Number;r>0;Z1: Line;Z2: Line;TangentOfPoint(P, C1) = Z1;TangentOfPoint(P, C2) = Z2;M: Point;TangentPoint(Z1, C1) = M;N: Point;TangentPoint(Z2, C2) = N;Min(Abs(LineSegmentOf(P, M))^2 - Abs(LineSegmentOf(P, N))^2) = 58", "coords": {} }, { "index": 907, "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/dev/parabola/problem_0907.png", "answer": "2", "fact_expr": "C: Parabola;G: Line;B: Point;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C)=F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = P", "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/dev/hyperbola/problem_0908.png", "answer": "(1, 2] + [3, 6)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);P: Point;PointOnCurve(P, RightPart(G));Distance(P, LeftFocus(G)) = 6*Distance(P, RightDirectrix(G))", "coords": {} }, { "index": 909, "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/dev/parabola/problem_0909.png", "answer": "x^2+(y-1)^2=3", "fact_expr": "G: Parabola;H: Circle;I: Line;Expression(G) = (x^2 = 4*y);Expression(I) = (y = x + 3);Center(H)=Focus(G);Length(InterceptChord(I,H))=2", "coords": {} }, { "index": 910, "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/dev/ellipse/problem_0910.png", "answer": "2", "fact_expr": "G: Ellipse;t: Number;t>1;Expression(G) = (y^2 + x^2/t = 1);F:Point;A:Point;B:Point;OneOf(Focus(G))=F;Endpoint(MinorAxis(G))={A,B};Area(TriangleOf(F,A,B))=1", "coords": {} }, { "index": 911, "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/dev/ellipse/problem_0911.png", "answer": "(-2/3, 1/3)", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 2*y^2 = 4);Expression(H) = (y = x + 1)", "coords": {} }, { "index": 912, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0912.png", "answer": "3/2", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;M: Point;B: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;PointOnCurve(F, l);Intersection(l,C)={A,B};Intersection(l,Directrix(C))= M;VectorOf(A, F) = VectorOf(F, M)", "coords": {} }, { "index": 913, "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/dev/parabola/problem_0913.png", "answer": "2", "fact_expr": "G: Line;C: Curve;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 2*x);Inclination(G) = ApplyUnit(60, degree);PointOnCurve(F, G);Intersection(G, C) = {A, B};Quadrant(A) = 1;Negation(A = B);Focus(C) = F", "coords": {} }, { "index": 914, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0914.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;O: Origin;F: Point;A:Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;IsDiameter(LineSegmentOf(O,F),H);Intersection(OneOf(Asymptote(G)),H)=A;Negation(A=O);IsPerpendicular(LineSegmentOf(A,MidPoint(LineSegmentOf(O,F))),LineSegmentOf(O,F));Eccentricity(G) = e", "coords": {} }, { "index": 915, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0915.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;N: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;PointOnCurve(M,LineSegmentOf(M,N));IsPerpendicular(LineSegmentOf(M,N),Directrix(G));FootPoint(LineSegmentOf(M,N),Directrix(G))=N", "coords": {} }, { "index": 916, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 917, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0917.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: LineSegment;Expression(G) = (y^2 = 2*p*x);Coordinate(Z) = (3, 1);Z: Point;MidPoint(H) = Z;IsChordOf(H,G);Slope(OverlappingLine(H))=2", "coords": { "Z": [ 3.0, 1.0 ] } }, { "index": 918, "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/dev/hyperbola/problem_0918.png", "answer": "y^2/8 - x^2/4 = 1", "fact_expr": "C1: Hyperbola;C2: Hyperbola;H: Point;Expression(C1) = (x^2/4 - y^2/8 = 1);Coordinate(H) = (2, 4);Asymptote(C1)=Asymptote(C2);PointOnCurve(H,C2)", "coords": { "H": [ 2.0, 4.0 ] } }, { "index": 919, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0919.png", "answer": "24*sqrt(13)/13", "fact_expr": "G: Ellipse;H: Line;Q: Point;Expression(G) = (x^2/4 + y^2/7 = 1);Expression(H) = (3*x - 2*y - 16 = 0);PointOnCurve(Q, G)", "coords": {} }, { "index": 920, "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/dev/parabola/problem_0920.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2 + (x - 2)^2 = 4);C:Line;PointOnCurve(Focus(G),C)", "coords": {} }, { "index": 921, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 922, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 923, "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/dev/hyperbola/problem_0923.png", "answer": "7", "fact_expr": "G: Hyperbola;P: Point;Q: Point;F1: Point;F2:Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(Q) = (-2, 3);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G))", "coords": { "Q": [ -2.0, 3.0 ] } }, { "index": 924, "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/dev/parabola/problem_0924.png", "answer": "(y - x^2)*(1 + 4*x^2) = 1", "fact_expr": "C: Parabola;B: Point;A: Point;P1: Point;P2: Point;M: Point;Expression(C) = (y = x^2);Length(LineSegmentOf(A, B)) = 2;Endpoint(LineSegmentOf(A, B)) = {P1, P2};PointOnCurve(P1, C);PointOnCurve(P2, C);MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 925, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/ellipse/problem_0926.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/2 + x^2/a^2 = 1);a: Number;a>sqrt(2);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H) = True;Slope(H) = -2;Intersection(H, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P;P: Point;O: Origin;Slope(LineOf(O, P)) = 1/4", "coords": {} }, { "index": 927, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 928, "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/dev/parabola/problem_0928.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;B: Point;A: Point;M: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (0, 1);PointOnCurve(P, Directrix(C));L1:Line;L2:Line;TangentOfPoint(P, C) = {L1,L2};TangentPoint(L1,C)=A;TangentPoint(L2,C)=B", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 929, "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/dev/parabola/problem_0929.png", "answer": "{((x-1)^2+y^2=2),((x+1)^2+y^2=2)}", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;Intersection(Directrix(G), yAxis) = Q;Q: Point;PointOnCurve(Q, L) = True;L: Line;TangentPoint(L, G) = P;P: Point", "coords": {} }, { "index": 930, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 931, "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/dev/hyperbola/problem_0931.png", "answer": "1", "fact_expr": "C: Hyperbola;F2: Point;Expression(C) = (x^2/4 - y^2 = 1);RightFocus(C) = F2", "coords": {} }, { "index": 932, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 933, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 9.0, "b": 5.0, "major_axis": "x", "x_coef": 81.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_0933.png", "answer": "12", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/81 + y^2/25 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 934, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 5.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/parabola/problem_0934.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 20*x)", "coords": {} }, { "index": 935, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 936, "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/dev/parabola/problem_0936.png", "answer": "x^2-y^2/3=1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;RightFocus(G) = Focus(H);Eccentricity(G) = 2", "coords": {} }, { "index": 937, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 938, "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/dev/hyperbola/problem_0938.png", "answer": "14", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2/4 - y^2/5 = 1);Coordinate(A) = (0, 4);LeftFocus(C)=F;PointOnCurve(P, RightPart(C))", "coords": { "A": [ 0.0, 4.0 ] } }, { "index": 939, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 940, "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/dev/hyperbola/problem_0940.png", "answer": "9*sqrt(3)/16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/3 = 1);P: Point;PointOnCurve(P, RightPart(G));Z1: Line;Z2: Line;K1: Line;K2: Line;Asymptote(G) = {K1, K2};PointOnCurve(P, Z1);PointOnCurve(P, Z2);IsPerpendicular(K1, Z1);IsPerpendicular(K2, Z2);M: Point;N: Point;FootPoint(K1, Z1) = M;FootPoint(K2, Z2) = N", "coords": {} }, { "index": 941, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 942, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0942.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(H) = (y^2 = a*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 943, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0943.png", "answer": "(1 + sqrt(5))/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;c: Number;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);RightFocus(C) = F;OneOf(Endpoint(ImageinaryAxis(C))) = B;IsPerpendicular(LineOf(B, F), OneOf(Asymptote(C)))", "coords": {} }, { "index": 944, "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/dev/hyperbola/problem_0944.png", "answer": "sqrt(5)/5", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 945, "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/dev/ellipse/problem_0945.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;O: Origin;Center(G) = O;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;UpperVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), xAxis);IsParallel(LineSegmentOf(P, F2), LineSegmentOf(A, B))", "coords": {} }, { "index": 946, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 947, "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/dev/hyperbola/problem_0947.png", "answer": "x^2+(y-5)^2=16", "fact_expr": "G: Hyperbola;H: Circle;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (0, 5);Center(H)=A;IsTangent(Asymptote(G),H)", "coords": { "A": [ 0.0, 5.0 ] } }, { "index": 948, "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/dev/ellipse/problem_0948.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;Distance(P, F1) = 3;OneOf(Focus(G)) = F2;Negation(F1 = F2)", "coords": {} }, { "index": 949, "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/dev/parabola/problem_0949.png", "answer": "3/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 12*x);M: Point;Coordinate(M) = (-3, 4);PointOnCurve(Focus(C), L);Slope(L) = k;k: Number;L: Line;Intersection(L, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -3.0, 4.0 ] } }, { "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/dev/hyperbola/problem_0950.png", "answer": "m - a^2", "fact_expr": "H: Ellipse;Expression(H) = (y^2/n + x^2/m = 1);m: Number;n: Number;m > n;n > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(H) = {F1, F2};Focus(G) = {F1, F2};F1: Point;F2: Point;P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 951, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0951.png", "answer": "5", "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));O: Origin;DotProduct((VectorOf(O, F2) + VectorOf(O, P)),VectorOf(P, F2)) = 0;3*Abs(VectorOf(P, F1)) = 4*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 952, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0952.png", "answer": "y^2=8*x", "fact_expr": "C: Parabola;p: Number;G: Line;Q: Point;F: Point;P: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (y = 4);Focus(C) = F;Intersection(G, yAxis) = P;Intersection(G,C)=Q;Abs(LineSegmentOf(Q, F)) = 2*Abs(LineSegmentOf(P, Q))", "coords": {} }, { "index": 953, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0953.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;m>0;F: Point;Focus(G) = F;Z: Line;PointOnCurve(F, Z);Slope(Z) = 2*sqrt(2);A: Point;B: Point;Intersection(Z, G) = {A, B};H: Circle;IsDiameter(LineSegmentOf(A, B), H);l: Line;Directrix(G) = l;M: Point;Intersection(H, l) = M;Abs(LineSegmentOf(M, F)) = sqrt(2)", "coords": {} }, { "index": 954, "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/dev/ellipse/problem_0954.png", "answer": "3/8, 2/3", "fact_expr": "A: Point;Coordinate(A) = (2, 0);B: Point;Coordinate(B) = (0, 1);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;In(A, Vertex(G));H: Line;Expression(H) = (y = k*x);k: Number;k>0;Intersection(H, LineOf(A, B)) = D;D: Point;Intersection(H, G) = {E, F};E: Point;F: Point;VectorOf(E, D) = 6*VectorOf(D, F);Slope(H) = k;In(B, Vertex(G))", "coords": { "A": [ 2.0, 0.0 ], "B": [ 0.0, 1.0 ] } }, { "index": 955, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 956, "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/dev/hyperbola/problem_0956.png", "answer": "(-sqrt(15)/3,-1)", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 6);Expression(H) = (y = k*x + 2);NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 957, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 958, "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/dev/parabola/problem_0958.png", "answer": "3/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);P:Point;PointOnCurve(P,G);Distance(P,Focus(G))=2*Distance(P,yAxis)", "coords": {} }, { "index": 959, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0959.png", "answer": "y= pm*(sqrt(3)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/6 = 1)", "coords": {} }, { "index": 960, "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/dev/hyperbola/problem_0960.png", "answer": "x + pm*sqrt(2)*y = 0", "fact_expr": "C2: Hyperbola;C1: Ellipse;a: Number;b: Number;a > b;b > 0;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);Expression(C2) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C1)*Eccentricity(C2) = sqrt(3)/2", "coords": {} }, { "index": 961, "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/dev/hyperbola/problem_0961.png", "answer": "36", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/64 - y^2/36 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 962, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0962.png", "answer": "2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;M: Point;N: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = M;RightFocus(C) = F;PointOnCurve(LeftVertex(C),l);Slope(l)=1;Intersection(l, RightPart(C)) = N;Area(TriangleOf(M, N, F)) = (3/2)*b^2", "coords": {} }, { "index": 963, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 964, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 965, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0965.png", "answer": "sqrt(3)", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsParallel(H, OneOf(Asymptote(G)));H: Line;Intersection(H, G) = P;P: Point;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 966, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 967, "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/dev/hyperbola/problem_0967.png", "answer": "y=pm*(sqrt(7)/7)*x\n8", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/14 + y^2/2 = 1)", "coords": {} }, { "index": 968, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 969, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 970, "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/dev/ellipse/problem_0970.png", "answer": "sqrt(2)", "fact_expr": "C1: Curve;C2: Curve;P: Point;F1: Point;F2: Point;Expression(C1) = (x^2/6+y^2/2=1);Expression(C2) = (x^2/3-y^2=1);Focus(C1) = {F2,F1};OneOf(Intersection(C1,C2))=P", "coords": {} }, { "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/dev/hyperbola/problem_0971.png", "answer": "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;Coordinate(M)=(a,b);AngleOf(M,F1,F2)=ApplyUnit(30,degree)", "coords": {} }, { "index": 972, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 973, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/dev/hyperbola/problem_0973.png", "answer": "x^2/9 - y^2 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;M: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1,F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0;Abs(VectorOf(M, F1))*Abs(VectorOf(M, F2)) = 2", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 974, "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/dev/parabola/problem_0974.png", "answer": "{(1,1),(-1,1)}", "fact_expr": "C: Parabola;M: Point;F: Point;Expression(C) = (y = x^2);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 5/4", "coords": {} }, { "index": 975, "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/dev/ellipse/problem_0975.png", "answer": "3+sqrt(6)", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 9);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G);Negation(In(P,Vertex(G)));O: Origin;D: Point;MidPoint(LineSegmentOf(P, F1)) = D", "coords": {} }, { "index": 976, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 977, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0977.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 1)", "coords": {} }, { "index": 978, "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/dev/hyperbola/problem_0978.png", "answer": "x+pm*2*y=0", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = sqrt(5)/2", "coords": {} }, { "index": 979, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 980, "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/dev/hyperbola/problem_0980.png", "answer": "44", "fact_expr": "F: Point;C: Hyperbola;Expression(C) = (x^2/9 - y^2/16 = 1);LeftFocus(C) = F;Q: Point;P: Point;PointOnCurve(P, C);PointOnCurve(Q, C);Length(LineSegmentOf(P, Q)) = Length(ImageinaryAxis(C)) * 2;A: Point;Coordinate(A) = (5, 0);PointOnCurve(A, LineSegmentOf(P, Q))", "coords": { "A": [ 5.0, 0.0 ] } }, { "index": 981, "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/dev/hyperbola/problem_0981.png", "answer": "y^2/7 - x^2/14 = 1", "fact_expr": "G: Hyperbola;H: Point;Q: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (2, 3);PointOnCurve(H, Q);Asymptote(G) = Asymptote(Q)", "coords": { "H": [ 2.0, 3.0 ] } }, { "index": 982, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 983, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_0983.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;C: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F,H);Intersection(H,G) = {A, B};Abs(LineSegmentOf(A,F))=4;PointOnCurve(C,Directrix(G));VectorOf(C,B)=2*VectorOf(B,F)", "coords": {} }, { "index": 984, "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/dev/ellipse/problem_0984.png", "answer": "3", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;Expression(C) = (x^2/12 + y^2/3 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = 3", "coords": {} }, { "index": 985, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0985.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;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C) = True;AngleOf(P, F1, F2) = ApplyUnit(15, degree);Area(TriangleOf(P, F1, F2)) = b^2", "coords": {} }, { "index": 986, "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/dev/parabola/problem_0986.png", "answer": "y = -1/2", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2)", "coords": {} }, { "index": 987, "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/dev/hyperbola/problem_0987.png", "answer": "(-sqrt(2), sqrt(2))&Negation(m=pm*1)", "fact_expr": "G: Hyperbola;H: Line;m: Real;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = m*x + 1);NumIntersection(H, G) = 2", "coords": {} }, { "index": 988, "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/dev/ellipse/problem_0988.png", "answer": "x^2/32+y^2/8=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(3)/2;l: Line;Expression(l) = (y = x + 2);A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;Slope(LineOf(O, A)) + Slope(LineOf(O, B)) = 4", "coords": {} }, { "index": 989, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 990, "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/dev/ellipse/problem_0990.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))", "coords": {} }, { "index": 991, "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/dev/parabola/problem_0991.png", "answer": "y=-1/32", "fact_expr": "G: Parabola;Expression(G) = (y = 8*x^2)", "coords": {} }, { "index": 992, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 993, "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/dev/hyperbola/problem_0993.png", "answer": "(1/3, 1)", "fact_expr": "G: Hyperbola;H: Line;b: Number;A: Point;C: Point;B: Point;b>0;Expression(H) = (y = b + x);Expression(G) = (x^2 - y^2/4 = 1);Intersection(H,LeftPart(G))=A;Intersection(H,RightPart(G))=B;Intersection(H,yAxis)=C", "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 }, "output_path": "results/dev/parabola/problem_0995.png", "answer": "1/3", "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": 996, "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/dev/parabola/problem_0996.png", "answer": "3", "fact_expr": "M: Point;Coordinate(M) = (-1, 0);PointOnCurve(M, l) = True;l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};A: Point;B: Point;Between(A, M, B) = True;F: Point;Focus(C) = F;Area(TriangleOf(M, B, F)) = 4*Area(TriangleOf(M, A, F))", "coords": { "M": [ -1.0, 0.0 ] } }, { "index": 997, "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/dev/parabola/problem_0997.png", "answer": "sqrt(2)/2", "fact_expr": "Gamma: Curve;Expression(Gamma) = (x^2 = 8*y);F: Point;Focus(Gamma) = F;P: Point;PointOnCurve(P,Gamma) = True;A: Point;Coordinate(A) = (0, -2)", "coords": { "A": [ 0.0, -2.0 ] } }, { "index": 998, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0998.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B: Point;F: 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, xAxis);L1:Line;L2:Line;Intersection(L,L1)=A;PointOnCurve(B, L2);lambda:Number;lambda>0;VectorOf(O, A) = lambda*VectorOf(B, F);DotProduct(VectorOf(A, B), VectorOf(O, B)) = 0;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2)", "coords": {} }, { "index": 999, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_0999.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);a: Number;b: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*((sqrt(3)/3)*x));Distance(Vertex(G), Asymptote(G)) = 1", "coords": {} }, { "index": 1000, "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/dev/ellipse/problem_1000.png", "answer": "(x-5/2)^2+(y-5/2)^2=13/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);A: Point;B: Point;C: Point;RightVertex(G) = A;UpperVertex(G) = B;Coordinate(C) = (2, 5);H: Circle;PointOnCurve(A, H);PointOnCurve(B, H);PointOnCurve(C, H)", "coords": { "C": [ 2.0, 5.0 ] } }, { "index": 1001, "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/dev/parabola/problem_1001.png", "answer": "(x+1)^2+(y-1/2)^2=1", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y = x^2/2)&(x<0);PointOnCurve(Center(H), G);IsTangent(Directrix(G),H);IsTangent(yAxis,H)", "coords": {} }, { "index": 1002, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_1002.png", "answer": "y^2 = 8*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C);Coordinate(MidPoint(LineSegmentOf(A, F))) = (2, 2)", "coords": {} }, { "index": 1003, "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/dev/hyperbola/problem_1003.png", "answer": "sqrt(2)", "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, OneOf(Asymptote(C)));G: Circle;IsDiameter(LineSegmentOf(F1, F2), G);PointOnCurve(P, G)", "coords": {} }, { "index": 1004, "success": false, "error": "Unsupported or unparseable expression" }, { "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/dev/ellipse/problem_1005.png", "answer": "Preserve all mathematical expressions, LaTeX formulas, symbols, and equations EXACTLY. \nDo not add explanations, do not rewrite formulas, do not omit any content. \n\nText:", "fact_expr": "G: Ellipse;H: Line;m:Number;M:Point;m>0;Expression(H) = (y = x*(sqrt(2)/2));Expression(G) = (x^2/16 + y^2/m^2 = 1);OneOf(Intersection(G,H))=M;Projection(M,xAxis)=RightFocus(G)", "coords": {} }, { "index": 1006, "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/dev/parabola/problem_1006.png", "answer": "8", "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))) = 3", "coords": {} }, { "index": 1007, "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/dev/hyperbola/problem_1007.png", "answer": "24", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;Expression(G) = (x^2/16 - y^2/4 = 1);PointOnCurve(RightFocus(G),l);Intersection(l,RightPart(G))={A,B};Abs(LineSegmentOf(A, B)) = 4;F:Point;LeftFocus(G)=F", "coords": {} }, { "index": 1008, "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/dev/parabola/problem_1008.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 1009, "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/dev/ellipse/problem_1009.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 1010, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_1010.png", "answer": "8", "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^2/16 + y^2/12 = 1);Focus(C) = Focus(G);InterReciprocal(Eccentricity(C), Eccentricity(G));F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P,RightPart(C))", "coords": {} }, { "index": 1011, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.25 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_1011.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 1)", "coords": {} }, { "index": 1012, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_1012.png", "answer": "4", "fact_expr": "C1:Parabola;C2:Hyperbola;Expression(C1) = (y^2=2*p*x);p:Number;p>0;Expression(C2) = (x^2/3-y^2/b^2=1);F:Point;Focus(C1)=F;OneOf(Focus(C2))=F;Distance(F,OneOf(Asymptote(C2)))=1", "coords": {} }, { "index": 1013, "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/dev/hyperbola/problem_1013.png", "answer": "3*sqrt(2)/4", "fact_expr": "C: Hyperbola;P: Point;F: Point;O: Origin;Expression(C) = (x^2/4 - y^2/2 = 1);RightFocus(C) = F;PointOnCurve(P, OneOf(Asymptote(C)));Abs(LineSegmentOf(P, O)) = Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 1014, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/dev/parabola/problem_1014.png", "answer": "2", "fact_expr": "A: Point;B: Point;G: Parabola;p: Number;H: Line;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(45, degree);Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 1015, "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/dev/ellipse/problem_1015.png", "answer": "x^2/18+y^2/9=1", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (1, -1);Coordinate(F) = (3, 0);RightFocus(E) = F;PointOnCurve(F, G);Intersection(G, E) = {A, B}", "coords": { "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1016, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/hyperbola/problem_1016.png", "answer": "y=pm*(4/3)*x", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: 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;PointOnCurve(F1,l);Slope(l)=a/b;Intersection(l,RightPart(C))=P;Intersection(l,OneOf(Asymptote(C)))=M;VectorOf(P, M) = 3*VectorOf(M, F1)", "coords": {} }, { "index": 1017, "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/dev/hyperbola/problem_1017.png", "answer": "{pm*1,pm*sqrt(6)/2}", "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2 - y^2 = 2);Expression(H) = (y = k*x + 1);NumIntersection(H, G)=1", "coords": {} }, { "index": 1018, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1019, "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/dev/hyperbola/problem_1019.png", "answer": "x^2/16 - y^2/4 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;e:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F)=(2*sqrt(5),0);OneOf(Focus(G))=F;Eccentricity(G)=e;e=sqrt(5)/2", "coords": { "F": [ 4.47213595499958, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1020, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1021, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1022, "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/dev/parabola/problem_1022.png", "answer": "10", "fact_expr": "C: Parabola;P: Point;PointOnCurve(P, C);H: Line;PointOnCurve(P, H);IsPerpendicular(H, Directrix(C)) = True;M: Point;FootPoint(H, Directrix(C)) = M;F: Point;Focus(C) = F;Abs(LineSegmentOf(P, F)) = 5;Expression(C)=(y^2=4*x)", "coords": {} }, { "index": 1023, "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/dev/parabola/problem_1023.png", "answer": "13", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 12", "coords": {} }, { "index": 1024, "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/dev/ellipse/problem_1024.png", "answer": "[1/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;m:Number;n: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);Max(Abs(VectorOf(P,F1))*Abs(VectorOf(P,F2)))=m;Min(DotProduct(VectorOf(P, F1), VectorOf(P, F2))) = n;m >= 2*n", "coords": {} }, { "index": 1025, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.47213595499958, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 20.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/dev/ellipse/problem_1025.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/20 + y^2/8 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 1026, "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/dev/ellipse/problem_1026.png", "answer": "pm*sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;k: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = k*x);Eccentricity(G) = sqrt(2)/2;XCoordinate(OneOf(Intersection(H,G))) = b", "coords": {} }, { "index": 1027, "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/dev/parabola/problem_1027.png", "answer": "3/8", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y = 2*x^2);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 1028, "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/dev/ellipse/problem_1028.png", "answer": "{4,5}", "fact_expr": "G: Curve;n: Number;m: Number;F1: Point;F2: Point;P: Point;m>0;n>0;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P,G);Perimeter(TriangleOf(P,F1,F2))=16;Min(Distance(P,F1))=2", "coords": {} }, { "index": 1029, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1030, "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/dev/parabola/problem_1030.png", "answer": "1/2", "fact_expr": "P: Point;Coordinate(P) = (1, -1);C: Parabola;Expression(C) = (y = x^2/4);G: Line;PointOnCurve(Focus(C), G);k: Number;Slope(G) = k;A: Point;B: Point;Intersection(G, C) = {A, B};DotProduct(VectorOf(P, A), VectorOf(P, B)) = 0", "coords": { "P": [ 1.0, -1.0 ] } }, { "index": 1031, "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/dev/ellipse/problem_1031.png", "answer": "(-2,2)", "fact_expr": "G: Ellipse;H: Line;m: Real;Expression(G) = (x^2/3 + y^2/4 = 1);Expression(H) = (y = m);NumIntersection(H, G) = 2", "coords": {} }, { "index": 1032, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1033, "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/dev/hyperbola/problem_1033.png", "answer": "8*x-y-15=0", "fact_expr": "l: Line;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, l);G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 1034, "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/dev/ellipse/problem_1034.png", "answer": "36/25", "fact_expr": "G: Ellipse;H:Line;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,H);IsPerpendicular(H,MajorAxis(G));Intersection(H,G) = {A, B}", "coords": {} } ] }