id int32 1 6.98k | annotation stringclasses 132
values | source stringlengths 7 17 | problem_level int32 0 28 | problem_text_cn stringlengths 20 201 | problem_text_en stringlengths 58 424 | problem_img stringlengths 5 8 | construction_cdl listlengths 1 28 | text_cdl listlengths 0 16 | image_cdl listlengths 0 16 | goal_cdl stringlengths 8 131 | problem_answer stringclasses 906
values | theorem_seqs listlengths 0 28 | theorem_seqs_dag_json stringlengths 13 3.3k | image imagewidth (px) 48 1.6k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6,701 | XiaoxiaoJin_2023-06-04 | GeoQA-9732 | 6 | 如图所示，EG=GF，∠DOE=50°，⊙O的圆心为O，CD是圆O的直径。求∠FCD的大小。 | As shown in the diagram, EG=GF, ∠DOE=50°, O is the center of circle O, CD is the diameter of ⊙O. Find the measure of ∠FCD. | 6701.png | [
"Shape(EO,OC,OCE)",
"Shape(OE,EG,GO)",
"Shape(CO,OG,GF,FC)",
"Shape(CF,OFC)",
"Shape(DG,GE,OED)",
"Shape(FG,GD,ODF)",
"Collinear(COGD)",
"Collinear(EGF)",
"Cocircular(O,CEDF)"
] | [
"Equal(LengthOfLine(EG),LengthOfLine(GF))",
"Equal(MeasureOfAngle(DOE),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(CD,O)"
] | [] | Value(MeasureOfAngle(FCD)) | 25 | [
"arc_property_center_angle(1,OED,O)",
"arc_property_circumference_angle_external(1,ODF,C)",
"circle_property_chord_perpendicular_bisect_chord(2,O,OG,EF)",
"circle_property_chord_perpendicular_bisect_arc(1,OEF,OGD)",
"congruent_arc_judgment_length_equal(1,OED,ODF)",
"congruent_arc_property_measure_equal(1,... | {"START": ["arc_property_center_angle(1,OED,O)", "arc_property_circumference_angle_external(1,ODF,C)", "circle_property_chord_perpendicular_bisect_chord(2,O,OG,EF)"], "circle_property_chord_perpendicular_bisect_arc(1,OEF,OGD)": ["congruent_arc_judgment_length_equal(1,OED,ODF)"], "circle_property_chord_perpendicular_bis... | |
6,702 | XiaoxiaoJin_2023-06-04 | GeoQA-9734 | 6 | 如图所示，∠CAB=20°，O是圆O的圆心，圆O的直径为AB，DE垂直于AE。求∠DOA的大小。 | As shown in the diagram, ∠CAB=20°, O is the center of ⊙O, the diameter of circle O is AB, DE is perpendicular to AE. Find the measure of ∠DOA. | 6702.png | [
"Shape(CA,AO,OE,EC)",
"Shape(AC,OCA)",
"Shape(DO,OA,OAD)",
"Shape(CE,EB,OBC)",
"Shape(BE,ED,ODB)",
"Shape(DO,DE,EO)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(CAB),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"PerpendicularBetweenLine(DE,AE)"
] | [] | Value(MeasureOfAngle(DOA)) | 140 | [
"arc_property_center_angle(1,ODB,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)",
"congruent_arc_judgment_length_equal(1,ODB,OBC)",
"congruent_arc_property_measure_equal(1,ODB,OBC)",
"adjacent_complementary_angle(1,EOD,DOA)"
] | {"START": ["arc_property_center_angle(1,ODB,O)", "arc_property_circumference_angle_external(1,OBC,A)", "circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)", "adjacent_complementary_angle(1,EOD,DOA)"], "circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)": ["congruent_arc_judgment_length_equal(1,ODB,OBC)"],... | |
6,703 | XiaoxiaoJin_2023-06-04 | GeoQA-9735 | 7 | 如图所示，∠BCA=15°，O是圆O的圆心。求∠OAB的大小。 | As shown in the diagram, ∠BCA=15°, O is the center of circle O. Find the measure of ∠OAB. | 6703.png | [
"Shape(OA,AD,DO)",
"Shape(DA,AB,BD)",
"Shape(CD,DB,BC)",
"Shape(CB,OBC)",
"Shape(BA,OAB)",
"Shape(AO,OD,DC,OCA)",
"Collinear(ODB)",
"Collinear(ADC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCA),15)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OAB)) | 75 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"isosceles_triangle_property_angle_equal(1,OAB)",
"triangle_pro... | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,OAB)"], "isosceles_triangle_judgment_line_equal(1,OAB)": ["isosceles_triangle_property... | |
6,704 | XiaoxiaoJin_2023-06-04 | GeoQA-9736 | 4 | 如图所示，∠DBA=53°，O是圆O的圆心，AB是圆O的直径。求∠BCD的大小。 | As shown in the diagram, ∠DBA=53°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠BCD. | 6704.png | [
"Shape(BD,ODB)",
"Shape(DB,BO,OE,ED)",
"Shape(DE,EA,OAD)",
"Shape(AE,EC,CA)",
"Shape(AC,OCA)",
"Shape(BC,CE,EO,OB)",
"Shape(CB,OBC)",
"Collinear(BOEA)",
"Collinear(DEC)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(DBA),53)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BCD)) | 37 | [
"angle_addition(1,BCE,ECA)",
"arc_property_circumference_angle_external(1,OAD,B)",
"arc_property_circumference_angle_external(1,OAD,C)",
"diameter_of_circle_property_right_angle(1,BCA,O)"
] | {"START": ["angle_addition(1,BCE,ECA)", "arc_property_circumference_angle_external(1,OAD,B)", "arc_property_circumference_angle_external(1,OAD,C)", "diameter_of_circle_property_right_angle(1,BCA,O)"]} | |
6,705 | XiaoxiaoJin_2023-06-04 | GeoQA-9739 | 5 | 如图所示，∠BCA=30°，⊙O的半径为1，O是圆O的圆心。求直线AB的长度。 | As shown in the diagram, ∠BCA=30°, the radius of ⊙O is 1, O is the center of circle O. Find the length of line AB. | 6705.png | [
"Shape(OA,AD,DO)",
"Shape(DA,AB,BD)",
"Shape(CD,DB,BC)",
"Shape(CB,OBC)",
"Shape(BA,OAB)",
"Shape(AO,OD,DC,OCA)",
"Collinear(ODB)",
"Collinear(ADC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCA),30)",
"Equal(RadiusOfCircle(O),1)",
"IsCentreOfCircle(O,O)"
] | [] | Value(LengthOfLine(AB)) | 1 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"cosine_theorem(1,OAB)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "cosine_theorem(1,OAB)"]} | |
6,706 | XiaoxiaoJin_2023-06-04 | GeoQA-9741 | 9 | 如图所示，∠CPA=28°，⊙O的圆心为O，AB是⊙O的直径，AO垂直于CO。求∠ABQ的大小。 | As shown in the diagram, ∠CPA=28°, the center of ⊙O is O, the diameter of circle O is AB, AO is perpendicular to CO. Find the measure of ∠ABQ. | 6706.png | [
"Shape(QC,OCQ)",
"Shape(CQ,QD,DC)",
"Shape(CD,DB,OBC)",
"Shape(DO,OB,BD)",
"Shape(OD,DQ,QO)",
"Shape(AO,OQ,OQA)",
"Shape(QP,PA,OQA)",
"Shape(BO,OA,OAB)",
"Collinear(PAOB)",
"Collinear(PQC)",
"Collinear(QDB)",
"Collinear(CDO)",
"Cocircular(O,ABCQ)"
] | [
"Equal(MeasureOfAngle(CPA),28)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"PerpendicularBetweenLine(AO,CO)"
] | [] | Value(MeasureOfAngle(ABQ)) | 17 | [
"angle_addition(1,AOQ,QOD)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OQ,O)",
"arc_property_center_angle(1,OQA,O)",
"arc_property_circumference_angle_external(1,OQA,B)",
"isosceles_triangle_judgment_line_equal(1,OCQ)",
"isosceles_triangle_property_angle... | {"START": ["angle_addition(1,AOQ,QOD)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OQ,O)", "arc_property_center_angle(1,OQA,O)", "arc_property_circumference_angle_external(1,OQA,B)", "triangle_property_angle_sum(1,CQO)", "triangle_property_angle_sum(1,CPO)"], "isosceles_... | |
6,707 | XiaoxiaoJin_2023-06-04 | GeoQA-9742 | 6 | 如图所示，∠OCD=40°，O是圆O的圆心，AB是圆O的直径，CE⊥BE。求∠DBA的大小。 | As shown in the diagram, ∠OCD=40°, the center of ⊙O is O, AB is the diameter of circle O, CE is perpendicular to BE. Find the measure of ∠DBA. | 6707.png | [
"Shape(CE,EO,OC)",
"Shape(AE,EC,OCA)",
"Shape(CE,EA,OAB)",
"Shape(BO,OE,ED,DB)",
"Shape(CO,OB,OBC)",
"Shape(BD,ODB)",
"Collinear(AEOB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(OCD),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"PerpendicularBetweenLine(CE,BE)"
] | [] | Value(MeasureOfAngle(DBA)) | 25 | [
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_angle_external(1,OAD,B)",
"circle_property_chord_perpendicular_bisect_arc(1,OCD,OEA)",
"congruent_arc_judgment_length_equal(1,OAD,OCA)",
"congruent_arc_property_measure_equal(1,OAD,OCA)",
"triangle_property_angle_sum(1,CEO)"
] | {"START": ["arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OAD,B)", "circle_property_chord_perpendicular_bisect_arc(1,OCD,OEA)", "triangle_property_angle_sum(1,CEO)"], "circle_property_chord_perpendicular_bisect_arc(1,OCD,OEA)": ["congruent_arc_judgment_length_equal(1,OAD,OCA)"], "con... | |
6,708 | XiaoxiaoJin_2023-06-04 | GeoQA-9746 | 4 | 如图所示，∠ABC=50°，O是⊙O的圆心，AB是⊙O的直径。求∠CDB的大小。 | As shown in the diagram, ∠ABC=50°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CDB. | 6708.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OE,EC)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(ED,DB,BE)",
"Shape(BD,ODB)",
"Shape(DE,EO,OA,OAD)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(ABC),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 40 | [
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,OBC,D)",
"triangle_property_angle_sum(1,CAB)",
"diameter_of_circle_property_right_angle(1,BCA,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OBC,D)", "triangle_property_angle_sum(1,CAB)", "diameter_of_circle_property_right_angle(1,BCA,O)"]} | |
6,709 | XiaoxiaoJin_2023-06-04 | GeoQA-9749 | 4 | 如图所示，∠COA=70°，O是⊙O的圆心，⊙O的直径为AB，DA和CB是平行四边形DCBA的一组对边。求∠BAD的大小。 | As shown in the diagram, ∠COA=70°, O is the center of circle O, AB is the diameter of circle O, CD and BA are opposite sides of the ▱ DCBA. Find the measure of ∠BAD. | 6709.png | [
"Shape(AD,DC,OAC)",
"Shape(CO,OA,OAC)",
"Shape(OC,CB,BO)",
"Shape(BC,OCB)",
"Shape(AO,OB,OBA)",
"Collinear(AOB)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(COA),70)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"Parallelogram(DCBA)"
] | [] | Value(MeasureOfAngle(BAD)) | 145 | [
"parallelogram_property_opposite_angle_equal(1,DCBA)",
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,B)",
"parallel_property_ipsilateral_internal_angle(1,AB,DC)"
] | {"START": ["parallelogram_property_opposite_angle_equal(1,DCBA)", "arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,B)", "parallel_property_ipsilateral_internal_angle(1,AB,DC)"]} | |
6,710 | XiaoxiaoJin_2023-06-04 | GeoQA-9750 | 4 | 如图所示，∠ABD=57°，O是⊙O的圆心，圆O的直径为AB。求∠DCB的大小。 | As shown in the diagram, ∠ABD=57°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠DCB. | 6710.png | [
"Shape(AO,OE,ED,DA)",
"Shape(AD,ODA)",
"Shape(DE,EB,BD)",
"Shape(DB,OBD)",
"Shape(BE,EC,OCB)",
"Shape(BC,OCB)",
"Shape(CE,EO,OA,OAC)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(ABD),57)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DCB)) | 33 | [
"diameter_of_circle_property_right_angle(1,BDA,O)",
"triangle_property_angle_sum(1,ABD)",
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,OBD,C)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BDA,O)", "triangle_property_angle_sum(1,ABD)", "arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,OBD,C)"]} | |
6,711 | XiaoxiaoJin_2023-06-04 | GeoQA-9752 | 2 | 如图所示，∠COB=50°，O是⊙O的圆心，AB是⊙O的直径。求∠CDB的大小。 | As shown in the diagram, ∠COB=50°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠CDB. | 6711.png | [
"Shape(DB,BE,ED)",
"Shape(EC,CO,OE)",
"Shape(BD,ODB)",
"Shape(CE,EB,OBC)",
"Shape(AO,OC,OCA)",
"Shape(DE,EO,OA,OAD)",
"Collinear(BEOA)",
"Collinear(DEC)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(COB),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 25 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,D)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,D)"]} | |
6,712 | XiaoxiaoJin_2023-06-04 | GeoQA-9753 | 4 | 如图所示，∠ABC=54°，O是圆O的圆心，AB是⊙O的直径。求∠CDB的大小。 | As shown in the diagram, ∠ABC=54°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠CDB. | 6712.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OE,EC)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(ED,DB,BE)",
"Shape(BD,ODB)",
"Shape(DE,EO,OA,OAD)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(ABC),54)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 36 | [
"diameter_of_circle_property_right_angle(1,BCA,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,OBC,D)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OBC,D)", "triangle_property_angle_sum(1,ABC)"]} | |
6,713 | XiaoxiaoJin_2023-06-04 | GeoQA-9754 | 2 | 如图所示，∠BAC=40°，⊙O的圆心为O。求∠BOC的大小。 | As shown in the diagram, ∠BAC=40°, O is the center of circle O. Find the measure of ∠BOC. | 6713.png | [
"Shape(BO,OC,CB)",
"Shape(BC,OCB)",
"Shape(OB,BA,AC,CO)",
"Shape(AB,OBA)",
"Shape(CA,OAC)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(BAC),40)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOC)) | 80 | [
"arc_property_center_angle(1,OCB,O)",
"arc_property_circumference_angle_external(1,OCB,A)"
] | {"START": ["arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OCB,A)"]} | |
6,714 | XiaoxiaoJin_2023-06-04 | GeoQA-9755 | 6 | 如图所示，⌒OAB的长度与⌒OCA的长度相等，∠CAB=50°，O是圆O的圆心，圆O的直径为AB。求∠AEC的大小。 | As shown in the diagram, the length of ⌒OAB is equal to the length of ⌒OCA, ∠CAB=50°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠AEC. | 6714.png | [
"Shape(BA,OAB)",
"Shape(AB,BD,DA)",
"Shape(AD,DC,CA)",
"Shape(AC,OCA)",
"Shape(DE,EC,CD)",
"Shape(ED,DB,OBE)",
"Shape(CE,OEC)",
"Collinear(BDC)",
"Collinear(ADE)",
"Cocircular(O,ABEC)"
] | [
"Equal(LengthOfArc(OAB),LengthOfArc(OCA))",
"Equal(MeasureOfAngle(CAB),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(AEC)) | 65 | [
"congruent_arc_judgment_length_equal(1,OAB,OCA)",
"congruent_arc_property_measure_equal(1,OAB,OCA)",
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_circumference_angle_external(1,OCA,E)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["congruent_arc_judgment_length_equal(1,OAB,OCA)", "arc_property_circumference_angle_external(1,OAB,C)", "arc_property_circumference_angle_external(1,OCA,B)", "arc_property_circumference_angle_external(1,OCA,E)", "triangle_property_angle_sum(1,ABC)"], "congruent_arc_judgment_length_equal(1,OAB,OCA)": ["congru... | |
6,715 | XiaoxiaoJin_2023-06-04 | GeoQA-9756 | 2 | 如图所示，∠COA=110°，O是⊙O的圆心，AB是⊙O的直径。求∠CBA的大小。 | As shown in the diagram, ∠COA=110°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CBA. | 6715.png | [
"Shape(BA,AD,DB)",
"Shape(AB,OBA)",
"Shape(OD,DC,CO)",
"Shape(CD,DA,OAC)",
"Shape(BD,DO,OC,OCB)",
"Collinear(BDC)",
"Collinear(ADO)",
"Cocircular(O,BAC)"
] | [
"Equal(MeasureOfAngle(COA),110)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CBA)) | 55 | [
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,B)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,B)"]} | |
6,716 | XiaoxiaoJin_2023-06-04 | GeoQA-9759 | 2 | 如图所示，∠COB=80°，⊙O的圆心为O，AB是圆O的直径。求∠CAB的大小。 | As shown in the diagram, ∠COB=80°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠CAB. | 6716.png | [
"Shape(AB,BO,OC,CA)",
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Shape(CO,OB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(COB),80)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 40 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,717 | XiaoxiaoJin_2023-06-04 | GeoQA-9760 | 2 | 如图所示，∠BOA=60°，O是圆O的圆心，圆O的直径为AB。求∠BCA的大小。 | As shown in the diagram, ∠BOA=60°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠BCA. | 6717.png | [
"Shape(OA,AD,DO)",
"Shape(DA,AB,BD)",
"Shape(CD,DB,BC)",
"Shape(BA,OAB)",
"Shape(CB,OBC)",
"Shape(AO,OD,DC,OCA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BOA),60)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 30 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)"]} | |
6,718 | XiaoxiaoJin_2023-06-04 | GeoQA-9761 | 2 | 如图所示，∠COA=110°，O是圆O的圆心，AB是圆O的直径。求∠CBA的大小。 | As shown in the diagram, ∠COA=110°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CBA. | 6718.png | [
"Shape(BA,AO,OC,CB)",
"Shape(AB,OBA)",
"Shape(BC,OCB)",
"Shape(CO,OA,OAC)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(COA),110)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CBA)) | 55 | [
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,B)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,B)"]} | |
6,719 | XiaoxiaoJin_2023-06-04 | GeoQA-9762 | 6 | 如图所示，∠CBO=25°，O是⊙O的圆心，AO垂直于BO。求∠DAC的大小。 | As shown in the diagram, ∠CBO=25°, O is the center of ⊙O, AO⊥BO. Find the measure of ∠DAC. | 6719.png | [
"Shape(AC,CD,DO,OA)",
"Shape(CA,OAC)",
"Shape(BD,DC,OCB)",
"Shape(OD,DB,BO)",
"Shape(AO,OB,OBA)",
"Collinear(CDB)",
"Collinear(AOD)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(CBO),25)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(AO,BO)"
] | [] | Value(MeasureOfAngle(DAC)) | 20 | [
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_angle_external(1,OBA,C)",
"triangle_property_angle_sum(1,ACD)",
"triangle_property_angle_sum(1,ODB)",
"adjacent_complementary_angle(1,CDO,ODB)",
"adjacent_complementary_angle(1,AOB,BOD)"
] | {"START": ["arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_external(1,OBA,C)", "triangle_property_angle_sum(1,ACD)", "triangle_property_angle_sum(1,ODB)", "adjacent_complementary_angle(1,CDO,ODB)", "adjacent_complementary_angle(1,AOB,BOD)"]} | |
6,720 | XiaoxiaoJin_2023-06-04 | GeoQA-9764 | 4 | 如图所示，∠CAB=42°，∠DBC=Mul(2,MeasureOfAngle(CDB))°，A是圆A的圆心。求∠DAC的大小。 | As shown in the diagram, ∠CAB=42°, ∠DBC=Mul(2,MeasureOfAngle(CDB))°, the center of ⊙A is A. Find the measure of ∠DAC. | 6720.png | [
"Shape(AB,BE,EA)",
"Shape(AE,ED,DA)",
"Shape(BC,CE,EB)",
"Shape(CD,DE,EC)",
"Shape(CB,OBC)",
"Shape(DC,OCD)",
"Shape(BA,AD,ODB)",
"Collinear(AEC)",
"Collinear(BED)",
"Cocircular(A,BCD)"
] | [
"Equal(MeasureOfAngle(CAB),42)",
"Equal(MeasureOfAngle(DBC),Mul(2,MeasureOfAngle(CDB)))",
"IsCentreOfCircle(A,A)"
] | [] | Value(MeasureOfAngle(DAC)) | 84 | [
"arc_property_center_angle(1,ABC,A)",
"arc_property_center_angle(1,ACD,A)",
"arc_property_circumference_angle_external(1,ABC,D)",
"arc_property_circumference_angle_external(1,ACD,B)"
] | {"START": ["arc_property_center_angle(1,ABC,A)", "arc_property_center_angle(1,ACD,A)", "arc_property_circumference_angle_external(1,ABC,D)", "arc_property_circumference_angle_external(1,ACD,B)"]} | |
6,721 | XiaoxiaoJin_2023-06-04 | GeoQA-9765 | 2 | 如图所示，∠BPA=15°，⊙O的圆心为O。求∠BOA的大小。 | As shown in the diagram, ∠BPA=15°, the center of ⊙O is O. Find the measure of ∠BOA. | 6721.png | [
"Shape(OA,AC,CO)",
"Shape(PC,CB,BP)",
"Shape(BC,CA,OAB)",
"Shape(PB,OBP)",
"Shape(AO,OC,CP,OPA)",
"Collinear(ACP)",
"Collinear(OCB)",
"Cocircular(O,ABP)"
] | [
"Equal(MeasureOfAngle(BPA),15)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOA)) | 30 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,P)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,P)"]} | |
6,722 | XiaoxiaoJin_2023-06-04 | GeoQA-9766 | 2 | 如图所示，∠BAC=60°，O是圆O的圆心。求∠BOC的大小。 | As shown in the diagram, ∠BAC=60°, O is the center of ⊙O. Find the measure of ∠BOC. | 6722.png | [
"Shape(AB,OBA)",
"Shape(BA,AC,CO,OB)",
"Shape(CA,OAC)",
"Shape(BO,OC,OCB)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(BAC),60)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOC)) | 120 | [
"arc_property_center_angle(1,OCB,O)",
"arc_property_circumference_angle_external(1,OCB,A)"
] | {"START": ["arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OCB,A)"]} | |
6,723 | XiaoxiaoJin_2023-06-04 | GeoQA-9767 | 10 | 如图所示，∠ABE=50°，O是圆O的圆心，AB是圆O的直径，DN⊥AN。求∠DCA的大小。 | As shown in the diagram, ∠ABE=50°, O is the center of circle O, AB is the diameter of ⊙O, DN⊥AN. Find the measure of ∠DCA. | 6723.png | [
"Shape(AC,OCA)",
"Shape(CA,AM,MC)",
"Shape(MD,DN,NO,OM)",
"Shape(EN,NB,BE)",
"Shape(EB,OBE)",
"Shape(BN,ND,DB)",
"Shape(BD,ODB)",
"Shape(DM,MA,OAD)",
"Shape(CM,MO,ON,NE,OEC)",
"Collinear(AMONB)",
"Collinear(CMD)",
"Collinear(END)",
"Cocircular(O,ADBEC)"
] | [
"Equal(MeasureOfAngle(ABE),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"PerpendicularBetweenLine(DN,AN)"
] | [] | Value(MeasureOfAngle(DCA)) | 50 | [
"arc_property_circumference_angle_external(1,OAD,B)",
"arc_property_circumference_angle_external(1,OAD,C)",
"circle_property_chord_perpendicular_bisect_chord(1,O,ON,DE)",
"circle_property_chord_perpendicular_bisect_arc(1,ODE,ONB)",
"congruent_arc_judgment_length_equal(1,ODB,OBE)",
"congruent_arc_property_... | {"START": ["arc_property_circumference_angle_external(1,OAD,B)", "arc_property_circumference_angle_external(1,OAD,C)", "circle_property_chord_perpendicular_bisect_chord(1,O,ON,DE)", "circle_property_chord_perpendicular_bisect_arc(1,ODE,ONB)"], "circle_property_chord_perpendicular_bisect_arc(1,ODE,ONB)": ["congruent_arc... | |
6,724 | XiaoxiaoJin_2023-06-04 | GeoQA-9772 | 2 | 如图所示，∠CAB=40°，⊙O的圆心为O。求∠COB的大小。 | As shown in the diagram, ∠CAB=40°, O is the center of ⊙O. Find the measure of ∠COB. | 6724.png | [
"Shape(BA,OAB)",
"Shape(AB,BO,OC,CA)",
"Shape(AC,OCA)",
"Shape(OB,BC,CO)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),40)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(COB)) | 80 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,725 | XiaoxiaoJin_2023-06-04 | GeoQA-9773 | 2 | 如图所示，∠BOC=50°，O是圆O的圆心，AB是⊙O的直径。求∠BAC的大小。 | As shown in the diagram, ∠BOC=50°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠BAC. | 6725.png | [
"Shape(AC,CO,OA)",
"Shape(CA,OAC)",
"Shape(AO,OB,OBA)",
"Shape(BO,OC,OCB)",
"Collinear(AOB)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(BOC),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BAC)) | 25 | [
"arc_property_center_angle(1,OCB,O)",
"arc_property_circumference_angle_external(1,OCB,A)"
] | {"START": ["arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OCB,A)"]} | |
6,726 | XiaoxiaoJin_2023-06-04 | GeoQA-9774 | 8 | 如图所示，∠DAB=60°，O是圆O的圆心，AB是⊙O的直径，AD∥OC。求∠DAC的大小。 | As shown in the diagram, ∠DAB=60°, the center of ⊙O is O, the diameter of ⊙O is AB, AD∥OC. Find the measure of ∠DAC. | 6726.png | [
"Shape(AD,ODA)",
"Shape(DA,AC,OCD)",
"Shape(CO,OB,OBC)",
"Shape(AO,OC,CA)",
"Shape(BO,OA,OAB)",
"Collinear(AOB)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(DAB),60)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"ParallelBetweenLine(AD,OC)"
] | [] | Value(MeasureOfAngle(DAC)) | 30 | [
"parallel_property_corresponding_angle(2,AD,OC,B)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"isosceles_triangle_judgment_line_equal(1,OCA)",
"isosceles_... | {"START": ["parallel_property_corresponding_angle(2,AD,OC,B)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "parallel_property_alternate_interior_angle(1,AD,OC)"], "isoscele... | |
6,727 | XiaoxiaoJin_2023-06-04 | GeoQA-9775 | 5 | 如图所示，∠ABC=30°，圆C的圆心为C，圆C的直径为EF，BC⊥AC。求∠BED的大小。 | As shown in the diagram, ∠ABC=30°, C is the center of ⊙C, the diameter of circle C is EF, BC⊥AC. Find the measure of ∠BED. | 6727.png | [
"Shape(DB,BF,ODF)",
"Shape(FC,CG,GD,ODF)",
"Shape(DA,OAD)",
"Shape(AD,DG,GA)",
"Shape(GC,CE,EG)",
"Shape(AG,GE,OEA)",
"Shape(EC,CF,OFE)",
"Collinear(BDA)",
"Collinear(BFCE)",
"Collinear(AGC)",
"Collinear(DGE)",
"Cocircular(C,ADFE)"
] | [
"Equal(MeasureOfAngle(ABC),30)",
"IsCentreOfCircle(C,C)",
"IsDiameterOfCircle(EF,C)",
"PerpendicularBetweenLine(BC,AC)"
] | [] | Value(MeasureOfAngle(BED)) | 15 | [
"arc_property_center_angle(1,CEA,C)",
"arc_property_circumference_angle_external(1,CEA,D)",
"adjacent_complementary_angle(1,ADG,GDB)",
"adjacent_complementary_angle(1,BCG,GCE)",
"triangle_property_angle_sum(1,DBE)"
] | {"START": ["arc_property_center_angle(1,CEA,C)", "arc_property_circumference_angle_external(1,CEA,D)", "adjacent_complementary_angle(1,ADG,GDB)", "adjacent_complementary_angle(1,BCG,GCE)", "triangle_property_angle_sum(1,DBE)"]} | |
6,728 | XiaoxiaoJin_2023-06-04 | GeoQA-9776 | 7 | 如图所示，∠OAB=40°，O是圆O的圆心，⊙O的直径为AB。求∠BCA的大小。 | As shown in the diagram, ∠OAB=40°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠BCA. | 6728.png | [
"Shape(BA,OAB)",
"Shape(BC,CD,DB)",
"Shape(BD,DA,AB)",
"Shape(AD,DO,OA)",
"Shape(CB,OBC)",
"Shape(BA,OAB)",
"Shape(AO,OD,DC,OCA)",
"Collinear(CDA)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(OAB),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 50 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"isosceles_triangle_property_angle_equal(1,OAB)",
"triangle_pro... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)", "triangle_property_angle_sum(1,ABO)"], "isosceles_triangle_judgment_line_equal(1,OAB)": ["isosceles_triangle_property... | |
6,729 | XiaoxiaoJin_2023-06-04 | GeoQA-9777 | 2 | 如图所示，∠COB=100°，⊙O的圆心为O。求∠CAB的大小。 | As shown in the diagram, ∠COB=100°, O is the center of ⊙O. Find the measure of ∠CAB. | 6729.png | [
"Shape(BA,OAB)",
"Shape(AB,BO,OC,CA)",
"Shape(AC,OCA)",
"Shape(OB,BC,CO)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(COB),100)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 50 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,730 | XiaoxiaoJin_2023-06-04 | GeoQA-9779 | 2 | 如图所示，∠BOA=140°，圆O的圆心为O。求∠ACB的大小。 | As shown in the diagram, ∠BOA=140°, the center of ⊙O is O. Find the measure of ∠ACB. | 6730.png | [
"Shape(BC,OCB)",
"Shape(CA,OAC)",
"Shape(CB,BO,OA,AC)",
"Shape(AO,OB,OBA)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(BOA),140)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ACB)) | 110 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_internal(1,OAB,C)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_internal(1,OAB,C)"]} | |
6,731 | XiaoxiaoJin_2023-06-04 | GeoQA-9780 | 5 | 如图所示，∠CAB=25°，∠DEC=30°，O是⊙O的圆心。求∠DOB的大小。 | As shown in the diagram, ∠CAB=25°, ∠DEC=30°, the center of circle O is O. Find the measure of ∠DOB. | 6731.png | [
"Shape(BA,OAB)",
"Shape(AB,BF,FA)",
"Shape(AF,FE,OEA)",
"Shape(FB,BM,MF)",
"Shape(CM,MB,OBC)",
"Shape(OM,MC,CN,NO)",
"Shape(DN,NC,OCD)",
"Shape(EN,ND,DE)",
"Shape(ED,ODE)",
"Shape(EF,FM,MO,ON,NE)",
"Collinear(AFMC)",
"Collinear(BMO)",
"Collinear(DNO)",
"Collinear(ENC)",
"Collinear(BFE)",... | [
"Equal(MeasureOfAngle(CAB),25)",
"Equal(MeasureOfAngle(DEC),30)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(DOB)) | 110 | [
"angle_addition(1,DEN,NEF)",
"arc_property_center_angle(1,OBD,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,OBC,E)",
"arc_property_circumference_angle_external(1,OBD,E)"
] | {"START": ["angle_addition(1,DEN,NEF)", "arc_property_center_angle(1,OBD,O)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OBC,E)", "arc_property_circumference_angle_external(1,OBD,E)"]} | |
6,732 | XiaoxiaoJin_2023-06-04 | GeoQA-9781 | 4 | 如图所示，OA=1，∠AOB=90°，∠BCO=60°，P是⊙P的圆心。求直线OB的长度。 | As shown in the diagram, OA=1, ∠AOB=90°, ∠BCO=60°, the center of circle P is P. Find the length of line OB. | 6732.png | [
"Shape(OA,PAO)",
"Shape(AD,DC,PCA)",
"Shape(CD,DB,BC)",
"Shape(CB,PBC)",
"Shape(BO,POB)",
"Shape(AO,OD,DA)",
"Shape(DO,OB,BD)",
"Collinear(ADB)",
"Collinear(CDO)",
"Cocircular(P,AOBC)"
] | [
"Equal(LengthOfLine(OA),1)",
"Equal(MeasureOfAngle(AOB),90)",
"Equal(MeasureOfAngle(BCO),60)",
"IsCentreOfCircle(P,P)"
] | [] | Value(LengthOfLine(OB)) | sqrt(3) | [
"arc_property_circumference_angle_external(1,POB,A)",
"arc_property_circumference_angle_external(1,POB,C)",
"triangle_property_angle_sum(1,AOB)",
"sine_theorem(1,OBA)"
] | {"START": ["arc_property_circumference_angle_external(1,POB,A)", "arc_property_circumference_angle_external(1,POB,C)", "triangle_property_angle_sum(1,AOB)", "sine_theorem(1,OBA)"]} | |
6,733 | XiaoxiaoJin_2023-06-04 | GeoQA-9784 | 5 | 如图所示，∠CAD=40°，圆O的圆心为O，BG垂直于OG。求∠DOB的大小。 | As shown in the diagram, ∠CAD=40°, O is the center of ⊙O, BG⊥OG. Find the measure of ∠DOB. | 6733.png | [
"Shape(AE,EO,OC,CA)",
"Shape(AC,OCA)",
"Shape(CO,OG,GC)",
"Shape(OE,EF,FG,GO)",
"Shape(EB,BF,FE)",
"Shape(BE,EA,OAB)",
"Shape(DF,FB,OBD)",
"Shape(FD,DG,GF)",
"Shape(CG,GD,ODC)",
"Collinear(AEFD)",
"Collinear(BEO)",
"Collinear(BFGC)",
"Collinear(OGD)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(CAD),40)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(BG,OG)"
] | [] | Value(MeasureOfAngle(DOB)) | 80 | [
"arc_property_center_angle(1,OBD,O)",
"arc_property_circumference_angle_external(1,ODC,A)",
"circle_property_chord_perpendicular_bisect_arc(1,OBC,OGD)",
"congruent_arc_judgment_length_equal(1,OBD,ODC)",
"congruent_arc_property_measure_equal(1,OBD,ODC)"
] | {"START": ["arc_property_center_angle(1,OBD,O)", "arc_property_circumference_angle_external(1,ODC,A)", "circle_property_chord_perpendicular_bisect_arc(1,OBC,OGD)"], "circle_property_chord_perpendicular_bisect_arc(1,OBC,OGD)": ["congruent_arc_judgment_length_equal(1,OBD,ODC)"], "congruent_arc_judgment_length_equal(1,OBD... | |
6,734 | XiaoxiaoJin_2023-06-04 | GeoQA-9786 | 3 | 如图所示，∠ABO=55°，圆O的圆心为O，BD垂直于AD。求∠COB的大小。 | As shown in the diagram, ∠ABO=55°, the center of circle O is O, BD⊥AD. Find the measure of ∠COB. | 6734.png | [
"Shape(BA,OAB)",
"Shape(AB,BD,DA)",
"Shape(OD,DC,CO)",
"Shape(CD,DB,OBC)",
"Shape(AD,DO,OC,OCA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABO),55)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(BD,AD)"
] | [] | Value(MeasureOfAngle(COB)) | 70 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"triangle_property_angle_sum(1,ABD)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "triangle_property_angle_sum(1,ABD)"]} | |
6,735 | XiaoxiaoJin_2023-06-04 | GeoQA-9787 | 7 | 如图所示，∠OAB=50°，圆O的圆心为O。求∠BCA的大小。 | As shown in the diagram, ∠OAB=50°, O is the center of circle O. Find the measure of ∠BCA. | 6735.png | [
"Shape(OA,AD,DO)",
"Shape(DA,AB,BD)",
"Shape(CD,DB,BC)",
"Shape(BA,OAB)",
"Shape(CB,OBC)",
"Shape(AO,OD,DC,OCA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(OAB),50)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 40 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"isosceles_triangle_property_angle_equal(1,OAB)",
"triangle_pro... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)", "triangle_property_angle_sum(1,OAB)"], "isosceles_triangle_judgment_line_equal(1,OAB)": ["isosceles_triangle_property... | |
6,736 | XiaoxiaoJin_2023-06-04 | GeoQA-9789 | 5 | 如图所示，弧OAB的长度与⌒OCA的长度相等，∠BCA=75°，O是圆O的圆心。求∠CAB的大小。 | As shown in the diagram, the length of ⌒OAB is equal to the length of arc OCA, ∠BCA=75°, the center of ⊙O is O. Find the measure of ∠CAB. | 6736.png | [
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Shape(AB,BC,CA)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(LengthOfArc(OAB),LengthOfArc(OCA))",
"Equal(MeasureOfAngle(BCA),75)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 30 | [
"congruent_arc_judgment_length_equal(1,OAB,OCA)",
"congruent_arc_property_chord_equal(1,OAB,OCA)",
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["congruent_arc_judgment_length_equal(1,OAB,OCA)", "triangle_property_angle_sum(1,ABC)"], "congruent_arc_judgment_length_equal(1,OAB,OCA)": ["congruent_arc_property_chord_equal(1,OAB,OCA)"], "congruent_arc_property_chord_equal(1,OAB,OCA)": ["isosceles_triangle_judgment_line_equal(1,ABC)"], "isosceles_triangle... | |
6,737 | XiaoxiaoJin_2023-06-04 | GeoQA-9790 | 3 | 如图所示，AB=10，AC=6，O是圆O的圆心，AB是圆O的直径。求直线BC的长度。 | As shown in the diagram, AB=10, AC=6, O is the center of circle O, the diameter of circle O is AB. Find the length of line BC. | 6737.png | [
"Shape(BC,OCB)",
"Shape(CB,BO,OA,AC)",
"Shape(CA,OAC)",
"Shape(AO,OB,OBA)",
"Collinear(AOB)",
"Cocircular(O,ACB)"
] | [
"Equal(LengthOfLine(AB),10)",
"Equal(LengthOfLine(AC),6)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(LengthOfLine(BC)) | 8 | [
"diameter_of_circle_property_right_angle(1,ACB,O)",
"right_triangle_judgment_angle(1,ACB)",
"right_triangle_property_pythagorean(1,ACB)"
] | {"START": ["diameter_of_circle_property_right_angle(1,ACB,O)"], "diameter_of_circle_property_right_angle(1,ACB,O)": ["right_triangle_judgment_angle(1,ACB)"], "right_triangle_judgment_angle(1,ACB)": ["right_triangle_property_pythagorean(1,ACB)"]} | |
6,738 | XiaoxiaoJin_2023-06-04 | GeoQA-9793 | 2 | 如图所示，∠BOA=38°，圆O的圆心为O。求∠BCA的大小。 | As shown in the diagram, ∠BOA=38°, the center of ⊙O is O. Find the measure of ∠BCA. | 6738.png | [
"Shape(AO,OD,DA)",
"Shape(DB,BC,CD)",
"Shape(BD,DA,OAB)",
"Shape(CB,OBC)",
"Shape(AO,OD,DC,OCA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BOA),38)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 19 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)"]} | |
6,739 | XiaoxiaoJin_2023-06-04 | GeoQA-9794 | 4 | 如图所示，∠ADC=35°，圆O的圆心为O，圆O的直径为BC。求∠BCA的大小。 | As shown in the diagram, ∠ADC=35°, O is the center of circle O, the diameter of circle O is BC. Find the measure of ∠BCA. | 6739.png | [
"Shape(BA,OAB)",
"Shape(AB,BO,OE,EA)",
"Shape(AE,EC,CA)",
"Shape(AC,OCA)",
"Shape(CE,ED,DC)",
"Shape(CD,ODC)",
"Shape(DE,EO,OB,OBD)",
"Collinear(BOEC)",
"Collinear(AED)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(ADC),35)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BC,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 55 | [
"diameter_of_circle_property_right_angle(1,CAB,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_circumference_angle_external(1,OCA,D)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["diameter_of_circle_property_right_angle(1,CAB,O)", "arc_property_circumference_angle_external(1,OCA,B)", "arc_property_circumference_angle_external(1,OCA,D)", "triangle_property_angle_sum(1,ABC)"]} | |
6,740 | XiaoxiaoJin_2023-06-04 | GeoQA-9798 | 4 | 如图所示，∠CDA=65°，O是圆O的圆心，圆O的直径为AB，CE⊥AE。求∠OCD的大小。 | As shown in the diagram, ∠CDA=65°, the center of circle O is O, the diameter of ⊙O is AB, CE⊥AE. Find the measure of ∠OCD. | 6740.png | [
"Shape(AD,ODA)",
"Shape(DA,AO,OE,ED)",
"Shape(OC,CE,EO)",
"Shape(BE,EC,CB)",
"Shape(DE,EB,OBD)",
"Shape(BC,COB)",
"Shape(CO,OA,OAC)",
"Collinear(DEC)",
"Collinear(AOEB)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(CDA),65)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"PerpendicularBetweenLine(CE,AE)"
] | [] | Value(MeasureOfAngle(OCD)) | 40 | [
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,D)",
"adjacent_complementary_angle(1,BOC,COA)",
"triangle_property_angle_sum(1,OCE)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,D)", "adjacent_complementary_angle(1,BOC,COA)", "triangle_property_angle_sum(1,OCE)"]} | |
6,741 | XiaoxiaoJin_2023-06-04 | GeoQA-9799 | 12 | 如图所示，∠CDA=62°，O是⊙O的圆心，圆O的直径为AB。求∠ACO的大小。 | As shown in the diagram, ∠CDA=62°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠ACO. | 6741.png | [
"Shape(BC,OCB)",
"Shape(CB,BE,EC)",
"Shape(CE,EO,OC)",
"Shape(CO,OA,AC)",
"Shape(CA,OAC)",
"Shape(AO,OE,ED,DA)",
"Shape(DE,EB,OBD)",
"Shape(AD,ODA)",
"Collinear(BEOA)",
"Collinear(DEC)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(CDA),62)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ACO)) | 28 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"arc_property_center_angle(1,OCB,O)",
"arc_property_circumference_angle_external(1,OAC,B)",
"arc_property_circumference_angle_external(1,OCB,A)",
"arc_pro... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OAC,B)", "arc_property_circumference_angle_external(1,OCB,A)", "arc_property... | |
6,742 | XiaoxiaoJin_2023-06-04 | GeoQA-9800 | 2 | 如图所示，∠ABC=30°，O是圆O的圆心。求∠AOC的大小。 | As shown in the diagram, ∠ABC=30°, the center of circle O is O. Find the measure of ∠AOC. | 6742.png | [
"Shape(AO,OC,OCA)",
"Shape(CO,OB,BC)",
"Shape(CB,OBC)",
"Shape(BO,OA,OAB)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABC),30)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOC)) | 60 | [
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_angle_external(1,OCA,B)"
] | {"START": ["arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OCA,B)"]} | |
6,743 | XiaoxiaoJin_2023-06-04 | GeoQA-9801 | 6 | 如图所示，AC=2，AO=3/2，⊙O的圆心为O，⊙O的直径为AD。求直线DC的长度。 | As shown in the diagram, AC=2, AO=3/2, O is the center of ⊙O, AD is the diameter of circle O. Find the length of line DC. | 6743.png | [
"Shape(AB,BE,EO,OA)",
"Shape(AO,OE,EC,CA)",
"Shape(DE,EB,OBD)",
"Shape(CE,ED,DC)",
"Shape(CD,ODC)",
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Collinear(AOED)",
"Collinear(BEC)",
"Cocircular(O,ABDC)"
] | [
"Equal(LengthOfLine(AC),2)",
"Equal(LengthOfLine(AO),3/2)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AD,O)"
] | [] | Value(LengthOfLine(DC)) | sqrt(5) | [
"line_addition(1,AO,OD)",
"diameter_of_circle_property_right_angle(1,DCA,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"right_triangle_judgment_angle(1,DCA)",
"right_triangle_property_pythagorean(1,DCA)"
] | {"START": ["line_addition(1,AO,OD)", "diameter_of_circle_property_right_angle(1,DCA,O)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OD,O)"], "diameter_of_circle_property_right_angle(1,DCA,O)": ["right_triangle_judgment_angle(1,DCA)"], "right_triangle_judgment_angle(1,DCA... | |
6,744 | XiaoxiaoJin_2023-06-04 | GeoQA-9803 | 4 | 如图所示，∠ACD=42°，O是圆O的圆心，⊙O的直径为AB。求∠DAB的大小。 | As shown in the diagram, ∠ACD=42°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DAB. | 6744.png | [
"Shape(AD,ODA)",
"Shape(DA,AE,ED)",
"Shape(AC,CE,EA)",
"Shape(DE,EO,OB,BD)",
"Shape(DB,OBD)",
"Shape(CA,OAC)",
"Shape(BO,OE,EC,OCB)",
"Collinear(AEOB)",
"Collinear(DEC)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(ACD),42)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DAB)) | 48 | [
"arc_property_circumference_angle_external(1,ODA,C)",
"arc_property_circumference_angle_external(1,ODA,B)",
"diameter_of_circle_property_right_angle(1,BDA,O)",
"triangle_property_angle_sum(1,DAB)"
] | {"START": ["arc_property_circumference_angle_external(1,ODA,C)", "arc_property_circumference_angle_external(1,ODA,B)", "diameter_of_circle_property_right_angle(1,BDA,O)", "triangle_property_angle_sum(1,DAB)"]} | |
6,745 | XiaoxiaoJin_2023-06-04 | GeoQA-9804 | 2 | 如图所示，∠CAB=20°，圆O的圆心为O，圆O的直径为AB。求∠COB的大小。 | As shown in the diagram, ∠CAB=20°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠COB. | 6745.png | [
"Shape(AC,OCA)",
"Shape(AO,OC,CA)",
"Shape(CO,OB,OBC)",
"Shape(BO,OA,OAB)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(COB)) | 40 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,746 | XiaoxiaoJin_2023-06-04 | GeoQA-9805 | 2 | 如图所示，∠AOB=100°，圆O的圆心为O。求∠ACB的大小。 | As shown in the diagram, ∠AOB=100°, O is the center of ⊙O. Find the measure of ∠ACB. | 6746.png | [
"Shape(AO,OD,DA)",
"Shape(BD,DC,CB)",
"Shape(BC,OCB)",
"Shape(AD,DB,OBA)",
"Shape(CD,DO,OA,OAC)",
"Collinear(BDO)",
"Collinear(ADC)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(AOB),100)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ACB)) | 50 | [
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_angle_external(1,OBA,C)"
] | {"START": ["arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_external(1,OBA,C)"]} | |
6,747 | XiaoxiaoJin_2023-06-04 | GeoQA-9806 | 3 | 如图所示，∠CDB=30°，O是圆O的圆心，AB是⊙O的直径。求∠AOC的大小。 | As shown in the diagram, ∠CDB=30°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠AOC. | 6747.png | [
"Shape(BD,ODB)",
"Shape(DB,BE,ED)",
"Shape(EC,CO,OE)",
"Shape(DE,EO,OA,OAD)",
"Shape(CE,EB,OBC)",
"Shape(AO,OC,OCA)",
"Collinear(BEOA)",
"Collinear(DEC)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(CDB),30)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(AOC)) | 120 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,D)",
"adjacent_complementary_angle(1,AOC,COB)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,D)", "adjacent_complementary_angle(1,AOC,COB)"]} | |
6,748 | XiaoxiaoJin_2023-06-04 | GeoQA-9809 | 2 | 如图所示，∠CPB=60°，P是圆P的圆心，圆P的直径为AB。求∠CDB的大小。 | As shown in the diagram, ∠CPB=60°, the center of circle P is P, AB is the diameter of ⊙P. Find the measure of ∠CDB. | 6748.png | [
"Shape(DB,BE,ED)",
"Shape(DE,EP,PD)",
"Shape(DP,PA,AD)",
"Shape(AP,PC,CA)",
"Shape(PE,EC,CP)",
"Shape(CE,EB,BC)",
"Shape(BD,PDB)",
"Shape(CB,PBC)",
"Shape(AC,PCA)",
"Shape(DA,PAD)",
"Collinear(BEPA)",
"Collinear(DEC)",
"Cocircular(P,ADBC)"
] | [
"Equal(MeasureOfAngle(CPB),60)",
"IsCentreOfCircle(P,P)",
"IsDiameterOfCircle(AB,P)"
] | [] | Value(MeasureOfAngle(CDB)) | 30 | [
"arc_property_center_angle(1,PBC,P)",
"arc_property_circumference_angle_external(1,PBC,D)"
] | {"START": ["arc_property_center_angle(1,PBC,P)", "arc_property_circumference_angle_external(1,PBC,D)"]} | |
6,749 | XiaoxiaoJin_2023-06-04 | GeoQA-9810 | 3 | 如图所示，∠BOD=140°，⊙O的圆心为O，AB是⊙O的直径。求∠DCA的大小。 | As shown in the diagram, ∠BOD=140°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠DCA. | 6749.png | [
"Shape(CA,AE,EC)",
"Shape(AC,OCA)",
"Shape(CE,EO,OB,BC)",
"Shape(CB,OBC)",
"Shape(OE,ED,DO)",
"Shape(DE,EA,OAD)",
"Shape(BO,OD,ODB)",
"Collinear(AOB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(BOD),140)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DCA)) | 20 | [
"adjacent_complementary_angle(1,BOD,DOA)",
"arc_property_center_angle(1,OAD,O)",
"arc_property_circumference_angle_external(1,OAD,C)"
] | {"START": ["adjacent_complementary_angle(1,BOD,DOA)", "arc_property_center_angle(1,OAD,O)", "arc_property_circumference_angle_external(1,OAD,C)"]} | |
6,750 | XiaoxiaoJin_2023-06-04 | GeoQA-9814 | 3 | 如图所示，∠AOC=110°，圆O的圆心为O，⊙O的直径为AB。求∠CDB的大小。 | As shown in the diagram, ∠AOC=110°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠CDB. | 6750.png | [
"Shape(BD,ODB)",
"Shape(BE,ED,DB)",
"Shape(DE,EO,OA,OAD)",
"Shape(CO,OE,EC)",
"Shape(CE,EB,OBC)",
"Shape(AO,OC,OCA)",
"Collinear(BEOA)",
"Collinear(DEC)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(AOC),110)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 35 | [
"adjacent_complementary_angle(1,AOC,COB)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,D)"
] | {"START": ["adjacent_complementary_angle(1,AOC,COB)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,D)"]} | |
6,751 | YangLi_2023-06-04 | GeoQA-9816 | 3 | 如图所示，⌒OBA全等于⌒OAC，∠AOB=50°，O是圆O的圆心。求∠CDE的大小。 | As shown in the diagram, ⌒OBA is congruent to ⌒OAC, ∠AOB=50°, the center of ⊙O is O. Find the measure of ∠CDE. | 6751.png | [
"Shape(CF,FA,OAC)",
"Shape(AF,FG,GA)",
"Shape(AG,GB,OBA)",
"Shape(FC,CE,EF)",
"Shape(GF,FE,EO,OG)",
"Shape(BG,GO,OB)",
"Shape(DC,OCD)",
"Shape(EC,CD,DE)",
"Shape(BO,OE,ED,ODB)",
"Collinear(AGO)",
"Collinear(AFED)",
"Collinear(CFGB)",
"Collinear(CEO)",
"Cocircular(O,DBAC)"
] | [
"CongruentBetweenArc(OBA,OAC)",
"Equal(MeasureOfAngle(AOB),50)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CDE)) | 25 | [
"congruent_arc_property_measure_equal(1,OBA,OAC)",
"arc_property_circumference_angle_external(1,OAC,D)",
"arc_property_center_angle(1,OBA,O)"
] | {"START": ["congruent_arc_property_measure_equal(1,OBA,OAC)", "arc_property_circumference_angle_external(1,OAC,D)", "arc_property_center_angle(1,OBA,O)"]} | |
6,752 | YangLi_2023-06-04 | GeoQA-9819 | 10 | 如图所示，⌒OBC全等于⌒ODB，⊙O的直径为10，∠CAO=32°，圆O的圆心为O，AB是⊙O的直径。求∠BOD的大小。 | As shown in the diagram, ⌒OBC is congruent to ⌒ODB, the diameter of ⊙O is 10, ∠CAO=32°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BOD. | 6752.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OC)",
"Shape(CO,OB,OBC)",
"Shape(BO,OD,ODB)",
"Shape(DO,OA,OAD)",
"Collinear(AOB)",
"Cocircular(O,DBCA)"
] | [
"CongruentBetweenArc(OBC,ODB)",
"Equal(DiameterOfCircle(O),10)",
"Equal(MeasureOfAngle(CAO),32)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [
"IsCentreOfCircle(O,O)"
] | Value(MeasureOfAngle(BOD)) | 64 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,OCA)",
"isosceles_triangle_property_angle_equal(1,OCA)",
"triangle_property_angle_sum(1,OCA)",
"flat_angle(1,AOB)",
"angle_addition(1,AOC,COB)",
"arc_property_c... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OCA)", "flat_angle(1,AOB)", "angle_addition(1,AOC,COB)", "arc_property_center_angle(1,ODB,O)", "arc_property_center_angle(1,OBC,O)", "congruent_arc_property_measure_equal(1,OBC,O... | |
6,753 | YangLi_2023-06-04 | GeoQA-9820 | 4 | 如图所示，BC=12，∠EAC=30°，O是圆O的圆心，BD是圆O的直径。求直线BD的长度。 | As shown in the diagram, BC=12, ∠EAC=30°, O is the center of circle O, the diameter of circle O is BD. Find the length of line BD. | 6753.png | [
"Shape(CA,OAC)",
"Shape(BC,OCB)",
"Shape(AC,CF,FA)",
"Shape(FC,CE,EF)",
"Shape(EC,CB,BE)",
"Shape(EB,BO,OE)",
"Shape(DF,FE,EO,OD)",
"Shape(AF,FD,ODAs)",
"Shape(DO,OB,OBD)",
"Collinear(CEO)",
"Collinear(CFD)",
"Collinear(BEFA)",
"Collinear(BOD)",
"Cocircular(O,CBDA)"
] | [
"Equal(LengthOfLine(BC),12)",
"Equal(MeasureOfAngle(EAC),30)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BD,O)"
] | [] | Value(LengthOfLine(BD)) | 24 | [
"diameter_of_circle_property_right_angle(1,DCB,O)",
"arc_property_circumference_angle_external(1,OCB,D)",
"arc_property_circumference_angle_external(1,OCB,A)",
"sine_theorem(1,BDC)"
] | {"START": ["diameter_of_circle_property_right_angle(1,DCB,O)", "arc_property_circumference_angle_external(1,OCB,D)", "arc_property_circumference_angle_external(1,OCB,A)", "sine_theorem(1,BDC)"]} | |
6,754 | YangLi_2023-06-04 | GeoQA-9821 | 2 | 如图所示，∠CAB=48°，圆O的圆心为O。求∠COB的大小。 | As shown in the diagram, ∠CAB=48°, the center of ⊙O is O. Find the measure of ∠COB. | 6754.png | [
"Shape(CO,OB,OBC)",
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Shape(AB,BO,OC,CA)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),48)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(COB)) | 96 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,755 | YangLi_2023-06-04 | GeoQA-9825 | 6 | 如图所示，∠CAO=70°，O是⊙O的圆心，圆O的直径为AB。求∠BCO的大小。 | As shown in the diagram, ∠CAO=70°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠BCO. | 6755.png | [
"Shape(BO,OA,OAB)",
"Shape(AC,OCA)",
"Shape(CA,AO,OC)",
"Shape(CO,OB,BC)",
"Shape(CB,OBC)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAO),70)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BCO)) | 20 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"isosceles_triangle_judgment_line_equal(1,OBC)",
"isosceles_triangle_property_angle_equal(1,OBC)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property_angle_equal(1,OBC)"], "radius_of_circle... | |
6,756 | YangLi_2023-06-04 | GeoQA-9827 | 4 | 如图所示，∠BCE=20°，O是⊙O的圆心，圆O的直径为AB。求∠DBE的大小。 | As shown in the diagram, ∠BCE=20°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DBE. | 6756.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OE,EC)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(BD,ODB)",
"Shape(BE,ED,DB)",
"Shape(DE,EO,OA,OAD)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(BCE),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DBE)) | 70 | [
"angle_addition(1,BCE,ECA)",
"arc_property_circumference_angle_external(1,OAD,B)",
"arc_property_circumference_angle_external(1,OAD,C)",
"diameter_of_circle_property_right_angle(1,BCA,O)"
] | {"START": ["angle_addition(1,BCE,ECA)", "arc_property_circumference_angle_external(1,OAD,B)", "arc_property_circumference_angle_external(1,OAD,C)", "diameter_of_circle_property_right_angle(1,BCA,O)"]} | |
6,757 | YangLi_2023-06-04 | GeoQA-9829 | 2 | 如图所示，∠OBC=30°，O是圆O的圆心，AB是圆O的直径。求∠CAO的大小。 | As shown in the diagram, ∠OBC=30°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠CAO. | 6757.png | [
"Shape(CA,AO,OB,BC)",
"Shape(AC,OCA)",
"Shape(CB,OBC)",
"Shape(BO,OA,OAB)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(OBC),30)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CAO)) | 60 | [
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"]} | |
6,758 | YangLi_2023-06-04 | GeoQA-9830 | 10 | 如图所示，∠OCA=25°，⊙O的圆心为O，CD是⊙O的直径，OE⊥BE。求∠BOE的大小。 | As shown in the diagram, ∠OCA=25°, the center of circle O is O, CD is the diameter of circle O, OE⊥BE. Find the measure of ∠BOE. | 6758.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OC)",
"Shape(OA,AE,EO)",
"Shape(OE,EB,BO)",
"Shape(CO,OB,OBC)",
"Shape(DE,EA,OAD)",
"Shape(BE,ED,ODB)",
"Collinear(COED)",
"Collinear(AEB)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(OCA),25)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(CD,O)",
"PerpendicularBetweenLine(OE,BE)"
] | [] | Value(MeasureOfAngle(BOE)) | 50 | [
"arc_property_center_angle(1,OAD,O)",
"arc_property_circumference_angle_external(1,OAD,C)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"flat_angle(1,AEB)",
"angle_addition(1,AEO,OEB)",
"triangle_property_angle_sum(1,OAE)",
"isosceles_triangle_ju... | {"START": ["arc_property_center_angle(1,OAD,O)", "arc_property_circumference_angle_external(1,OAD,C)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "flat_angle(1,AEB)", "angle_addition(1,AEO,OEB)", "triangle_property_angle_sum(1,OAE)", "triangle_property_angle_sum(... | |
6,759 | YangLi_2023-06-04 | GeoQA-9835 | 7 | 如图所示，∠CAB=68°，O是圆O的圆心。求∠OBC的大小。 | As shown in the diagram, ∠CAB=68°, O is the center of circle O. Find the measure of ∠OBC. | 6759.png | [
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Shape(OB,BC,CO)",
"Shape(AB,BO,OC,CA)",
"Shape(CB,OBC)",
"Cocircular(O,BCA)"
] | [
"Equal(MeasureOfAngle(CAB),68)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OBC)) | 22 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"isosceles_triangle_judgment_line_equal(1,OBC)",
"isosceles_triangle_property_angle_equal(1,OBC)",
"triangle_pro... | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "triangle_property_angle_sum(1,OBC)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property... | |
6,760 | YangLi_2023-06-04 | GeoQA-9838 | 8 | 如图所示，∠OBC=40°，圆O的圆心为O。求∠BAC的大小。 | As shown in the diagram, ∠OBC=40°, O is the center of ⊙O. Find the measure of ∠BAC. | 6760.png | [
"Shape(OB,BC,CO)",
"Shape(CB,BA,AC)",
"Shape(CA,OAC)",
"Shape(AB,OBA)",
"Shape(BO,OC,OCB)",
"Cocircular(O,BAC)"
] | [
"Equal(MeasureOfAngle(OBC),40)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BAC)) | 130 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,OBC)",
"isosceles_triangle_property_angle_equal(1,OBC)",
"triangle_property_angle_sum(1,OBC)",
"round_angle(1,COB,BOC)",
"arc_property_center_angle(1,OCB,O)",
"... | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OBC)", "round_angle(1,COB,BOC)", "arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OCB,A)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["i... | |
6,761 | YangLi_2023-06-04 | GeoQA-9839 | 2 | 如图所示，∠ABO=30°，圆O的圆心为O，BC是⊙O的直径。求∠OCA的大小。 | As shown in the diagram, ∠ABO=30°, the center of ⊙O is O, BC is the diameter of circle O. Find the measure of ∠OCA. | 6761.png | [
"Shape(AB,BO,OC,CA)",
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Shape(CO,OB,OBC)",
"Collinear(BOC)",
"Cocircular(O,BCA)"
] | [
"Equal(MeasureOfAngle(ABO),30)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BC,O)"
] | [] | Value(MeasureOfAngle(OCA)) | 60 | [
"diameter_of_circle_property_right_angle(1,CAB,O)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["diameter_of_circle_property_right_angle(1,CAB,O)", "triangle_property_angle_sum(1,ABC)"]} | |
6,762 | YangLi_2023-06-04 | GeoQA-9840 | 2 | 如图所示，∠CAE=25°，⊙O的圆心为O。求∠EOB的大小。 | As shown in the diagram, ∠CAE=25°, the center of ⊙O is O. Find the measure of ∠EOB. | 6762.png | [
"Shape(AC,OCA)",
"Shape(CA,AE,EC)",
"Shape(CE,EB,OBC)",
"Shape(BE,EO,OB)",
"Shape(EA,AO,OE)",
"Shape(BO,OA,OAB)",
"Collinear(CEO)",
"Collinear(AEB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAE),25)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(EOB)) | 50 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"]} | |
6,763 | YangLi_2023-06-04 | GeoQA-9844 | 8 | 如图所示，∠CBA=55°，O是⊙O的圆心，AO平行于DC。求∠DOA的大小。 | As shown in the diagram, ∠CBA=55°, O is the center of ⊙O, AO∥DC. Find the measure of ∠DOA. | 6763.png | [
"Shape(BA,AO,OC,CB)",
"Shape(OD,DC,CO)",
"Shape(DO,OA,OAD)",
"Shape(AB,OBA)",
"Shape(BC,OCB)",
"Shape(CD,ODC)",
"Cocircular(O,ADCB)"
] | [
"Equal(MeasureOfAngle(CBA),55)",
"IsCentreOfCircle(O,O)",
"ParallelBetweenLine(AO,DC)"
] | [] | Value(MeasureOfAngle(DOA)) | 70 | [
"radius_of_circle_property_length_equal(1,OD,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,ODC)",
"isosceles_triangle_property_angle_equal(1,ODC)",
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,B)",
"parallel_pro... | {"START": ["radius_of_circle_property_length_equal(1,OD,O)", "radius_of_circle_property_length_equal(1,OC,O)", "arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,B)", "parallel_property_alternate_interior_angle(2,AO,DC)", "parallel_property_ipsilateral_internal_angle(1,CD,OA)"], "iso... | |
6,764 | YangLi_2023-06-04 | GeoQA-9846 | 4 | 如图所示，∠CAO=28°，圆O的圆心为O，圆O的直径为AB。求∠ADE的大小。 | As shown in the diagram, ∠CAO=28°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠ADE. | 6764.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OE,EC)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(DA,OAD)",
"Shape(EO,OA,AD,DE)",
"Shape(BE,ED,ODB)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(CAO),28)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ADE)) | 62 | [
"arc_property_circumference_angle_external(1,OCA,D)",
"arc_property_circumference_angle_external(1,OCA,B)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,D)", "arc_property_circumference_angle_external(1,OCA,B)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"]} | |
6,765 | YangLi_2023-06-04 | GeoQA-9850 | 8 | 如图所示，∠ABC=130°，O是圆O的圆心。求∠ACO的大小。 | As shown in the diagram, ∠ABC=130°, O is the center of circle O. Find the measure of ∠ACO. | 6765.png | [
"Shape(BA,OAB)",
"Shape(CB,OBC)",
"Shape(CA,AB,BC)",
"Shape(OA,AC,CO)",
"Shape(AO,OC,OCA)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABC),130)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ACO)) | 40 | [
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"round_angle(1,AOC,COA)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,OAC)",
"isosceles_triangle_property_angle_eq... | {"START": ["arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OCA,B)", "round_angle(1,AOC,COA)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OAC)"], "isosceles_triangle_judgment_line_equal(1,OAC)": ["i... | |
6,766 | YangLi_2023-06-04 | GeoQA-9851 | 2 | 如图所示，∠CBA=52°，O是⊙O的圆心。求∠COA的大小。 | As shown in the diagram, ∠CBA=52°, O is the center of ⊙O. Find the measure of ∠COA. | 6766.png | [
"Shape(BA,AO,OC,CB)",
"Shape(AB,OBA)",
"Shape(BC,OCB)",
"Shape(CO,OA,AC)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(CBA),52)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(COA)) | 104 | [
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,B)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,B)"]} | |
6,767 | YangLi_2023-06-04 | GeoQA-9853 | 4 | 如图所示，∠ADE=55°，O是⊙O的圆心，AB是⊙O的直径。求∠CAO的大小。 | As shown in the diagram, ∠ADE=55°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠CAO. | 6767.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OE,EC)",
"Shape(CE,EB,BC)",
"Shape(CB,OBC)",
"Shape(EO,OA,AD,DE)",
"Shape(BE,ED,ODB)",
"Shape(DA,OAD)",
"Collinear(CED)",
"Collinear(AOEB)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(ADE),55)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CAO)) | 35 | [
"arc_property_circumference_angle_external(1,OCA,D)",
"arc_property_circumference_angle_external(1,OCA,B)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,CAB)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,D)", "arc_property_circumference_angle_external(1,OCA,B)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,CAB)"]} | |
6,768 | YangLi_2023-06-04 | GeoQA-9857 | 11 | 如图所示，∠CBO=19°，∠OAC=19°，圆O的圆心为O。求∠AOB的大小。 | As shown in the diagram, ∠CBO=19°, ∠OAC=19°, the center of ⊙O is O. Find the measure of ∠AOB. | 6768.png | [
"Shape(CA,OAC)",
"Shape(BC,OCB)",
"Shape(AO,OB,OBA)",
"Shape(AC,CO,OA)",
"Shape(OC,CB,BO)",
"Cocircular(O,CBA)"
] | [
"Equal(MeasureOfAngle(CBO),19)",
"Equal(MeasureOfAngle(OAC),19)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOB)) | 76 | [
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"isosceles_triangle_judgment_line_equal(1,OAC)",
"isosceles_triangle_judgment_line_equal(1,OCB)",
"isosceles_triangle_property_angle_equal(1,OAC)",
"isosc... | {"START": ["radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OA,O)", "triangle_property_angle_sum(1,ACO)", "triangle_property_angle_sum(1,OCB)", "angle_addition(1,BOC,COA)", "round_angle(1,AOB,BOA)"], "isosceles_triangle_judgmen... | |
6,769 | YangLi_2023-06-04 | GeoQA-9858 | 2 | 如图所示，∠COE=70°，圆O的圆心为O，⊙O的直径为AB。求∠EDB的大小。 | As shown in the diagram, ∠COE=70°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠EDB. | 6769.png | [
"Shape(BD,ODB)",
"Shape(DB,BE,ED)",
"Shape(DE,EO,OA,OAD)",
"Shape(AO,OC,OCA)",
"Shape(OE,EC,CO)",
"Shape(CE,EB,OBC)",
"Collinear(BEOA)",
"Collinear(DEC)",
"Cocircular(O,BCAD)"
] | [
"Equal(MeasureOfAngle(COE),70)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(EDB)) | 35 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,D)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,D)"]} | |
6,770 | YangLi_2023-06-04 | GeoQA-9861 | 8 | 如图所示，BC=2，∠CAD=30°，圆O的圆心为O。求直线OB的长度。 | As shown in the diagram, BC=2, ∠CAD=30°, the center of ⊙O is O. Find the length of line OB. | 6770.png | [
"Shape(BO,OD,DA,OAB)",
"Shape(OB,BD,DO)",
"Shape(AD,DC,CA)",
"Shape(DB,BC,CD)",
"Shape(AC,OCA)",
"Shape(CB,OBC)",
"Collinear(ODC)",
"Collinear(BDA)",
"Cocircular(O,BCA)"
] | [
"Equal(LengthOfLine(BC),2)",
"Equal(MeasureOfAngle(CAD),30)",
"IsCentreOfCircle(O,O)"
] | [] | Value(LengthOfLine(OB)) | 2 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"isosceles_triangle_judgment_line_equal(1,OBC)",
"isosceles_triangle_property_angle_equal(1,OBC)",
"triangle_property_angle_sum(1,OBC)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_a... | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OBC)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)"], "arc_property_center_angle(1,OBC,O)": ["isosceles_triangle_judgment_angle_equa... | |
6,771 | YangLi_2023-06-04 | GeoQA-9866 | 4 | 如图所示，∠EDA=55°，O是⊙O的圆心，圆O的直径为AB。求∠EAC的大小。 | As shown in the diagram, ∠EDA=55°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠EAC. | 6771.png | [
"Shape(AD,ODA)",
"Shape(DA,AE,ED)",
"Shape(DE,EO,OB,OBD)",
"Shape(BO,OE,EC,CB)",
"Shape(EA,AC,CE)",
"Shape(CA,OAC)",
"Shape(BC,OCB)",
"Collinear(AEOB)",
"Collinear(DEC)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(EDA),55)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(EAC)) | 35 | [
"arc_property_circumference_angle_external(1,OAC,B)",
"arc_property_circumference_angle_external(1,OAC,D)",
"diameter_of_circle_property_right_angle(1,ACB,O)",
"triangle_property_angle_sum(1,CBA)"
] | {"START": ["arc_property_circumference_angle_external(1,OAC,B)", "arc_property_circumference_angle_external(1,OAC,D)", "diameter_of_circle_property_right_angle(1,ACB,O)", "triangle_property_angle_sum(1,CBA)"]} | |
6,772 | YangLi_2023-06-04 | GeoQA-9867 | 6 | 如图所示，DO=AD，∠DAF=22°，O是⊙O的圆心。求∠GFO的大小。 | As shown in the diagram, DO=AD, ∠DAF=22°, O is the center of ⊙O. Find the measure of ∠GFO. | 6772.png | [
"Shape(DA,AF,ODF)",
"Shape(FG,GD,ODF)",
"Shape(GF,FO,OG)",
"Shape(ED,DG,GE)",
"Shape(DE,OED)",
"Shape(EG,GO,OB,OBE)",
"Shape(BO,OF,OFB)",
"Collinear(DGO)",
"Collinear(EGF)",
"Collinear(ADE)",
"Collinear(AFOB)",
"Cocircular(O,FGED)"
] | [
"Equal(LengthOfLine(DO),LengthOfLine(AD))",
"Equal(MeasureOfAngle(DAF),22)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(GFO)) | 33 | [
"isosceles_triangle_judgment_line_equal(1,DAO)",
"isosceles_triangle_property_angle_equal(1,DAO)",
"arc_property_center_angle(1,ODF,O)",
"arc_property_circumference_angle_external(1,ODF,E)",
"triangle_property_angle_sum(1,EAF)",
"adjacent_complementary_angle(1,AFG,GFO)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,DAO)", "arc_property_center_angle(1,ODF,O)", "arc_property_circumference_angle_external(1,ODF,E)", "triangle_property_angle_sum(1,EAF)", "adjacent_complementary_angle(1,AFG,GFO)"], "isosceles_triangle_judgment_line_equal(1,DAO)": ["isosceles_triangle_property_angle_e... | |
6,773 | YangLi_2023-06-04 | GeoQA-9871 | 7 | 如图所示，∠OCA=45°，圆O的圆心为O。求∠ABC的大小。 | As shown in the diagram, ∠OCA=45°, O is the center of circle O. Find the measure of ∠ABC. | 6773.png | [
"Shape(NC,CO,OA,AB)",
"Shape(OC,CA,AO)",
"Shape(CB,OBC)",
"Shape(BA,OAB)",
"Shape(AC,OCA)",
"Cocircular(O,CAB)"
] | [
"Equal(MeasureOfAngle(OCA),45)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 45 | [
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"isosceles_triangle_judgment_line_equal(1,OCA)",
"isosceles_triangle_property_angle_equal(1,OCA)",
"triangle_property_angle_sum(1,OCA)",
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_a... | {"START": ["radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OA,O)", "triangle_property_angle_sum(1,OCA)", "arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OCA,B)"], "isosceles_triangle_judgment_line_equal(1,OCA)": ["isosceles_triangle_property... | |
6,774 | YangLi_2023-06-04 | GeoQA-9873 | 9 | 如图所示，∠OAE=50°，O是圆O的圆心，AB是圆O的直径，⊙O的切线为DC，ED⊥CD。求∠DCB的大小。 | As shown in the diagram, ∠OAE=50°, O is the center of circle O, the diameter of circle O is AB, DC is the tangent to ⊙O, ED is perpendicular to CD. Find the measure of ∠DCB. | 6774.png | [
"Shape(AO,OB,OBA)",
"Shape(BO,OC,CB)",
"Shape(BC,OCB)",
"Shape(CO,OA,AE,OEC)",
"Shape(ED,DC,OEC)",
"Shape(EA,OAE)",
"Collinear(AED)",
"Collinear(AOB)",
"Cocircular(O,AECB)"
] | [
"Equal(MeasureOfAngle(OAE),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(DC,O)",
"PerpendicularBetweenLine(ED,CD)"
] | [] | Value(MeasureOfAngle(DCB)) | 155 | [
"tangent_of_circle_property_perpendicular(2,DC,O,O)",
"parallel_judgment_ipsilateral_internal_angle(1,DA,CO)",
"parallel_property_corresponding_angle(1,OC,AD,B)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,O... | {"START": ["tangent_of_circle_property_perpendicular(2,DC,O,O)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,OCB)", "angle_addition(1,DCO,OCB)"], "isosceles_triangle_judgment_line_equal(1,OCB)": ["isosceles_triangle_property_angle_eq... | |
6,775 | YangLi_2023-06-04 | GeoQA-9874 | 13 | 如图所示，AB=2，AC=BC，OC=1，⊙O的圆心为O，⊙O的切线为PA，⊙O的切线为PB，OC⊥BC。求∠APB的大小。 | As shown in the diagram, AB=2, AC=BC, OC=1, O is the center of circle O, PA is the tangent to circle O, PB is the tangent to ⊙O, OC is perpendicular to BC. Find the measure of ∠APB. | 6775.png | [
"Shape(OA,AC,CO)",
"Shape(OC,CB,BO)",
"Shape(AO,OB,OBA)",
"Shape(BC,CA,OAB)",
"Shape(AP,PB,OAB)",
"Collinear(ACB)",
"Cocircular(O,AB)"
] | [
"Equal(LengthOfLine(AB),2)",
"Equal(LengthOfLine(AC),LengthOfLine(BC))",
"Equal(LengthOfLine(OC),1)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PA,O)",
"IsTangentOfCircle(PB,O)",
"PerpendicularBetweenLine(OC,BC)"
] | [] | Value(MeasureOfAngle(APB)) | 90 | [
"tangent_of_circle_property_perpendicular(1,PA,O,O)",
"tangent_of_circle_property_perpendicular(2,PB,O,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"line_addition(1,AC,CB)",
"adjacent_complementary_angle(1,ACO,OCB)",
"isosceles_triangle_judgmen... | {"START": ["tangent_of_circle_property_perpendicular(1,PA,O,O)", "tangent_of_circle_property_perpendicular(2,PB,O,O)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "line_addition(1,AC,CB)", "adjacent_complementary_angle(1,ACO,OCB)", "triangle_property_angle_sum(1,O... | |
6,776 | YangLi_2023-06-04 | GeoQA-9880 | 6 | 如图所示，∠ABO=55°，O是圆O的圆心，TA是⊙O的切线。求∠BAT的大小。 | As shown in the diagram, ∠ABO=55°, O is the center of ⊙O, TA is the tangent to circle O. Find the measure of ∠BAT. | 6776.png | [
"Shape(AO,OB,OBA)",
"Shape(OA,AB,BO)",
"Shape(BA,OAB)",
"Shape(TA,OAB)",
"Cocircular(O,AB)"
] | [
"Equal(MeasureOfAngle(ABO),55)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(TA,O)"
] | [] | Value(MeasureOfAngle(BAT)) | 35 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"isosceles_triangle_property_angle_equal(1,OAB)",
"tangent_of_circle_property_perpendicular(1,TA,O,O)",
"angle_addition(1,OAB,BAT)"
] | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "tangent_of_circle_property_perpendicular(1,TA,O,O)", "angle_addition(1,OAB,BAT)"], "isosceles_triangle_judgment_line_equal(1,OAB)": ["isosceles_triangle_property_angle_equal(1,OAB)"], "radius_of_circle_proper... | |
6,777 | YangLi_2023-06-04 | GeoQA-9881 | 4 | 如图所示，OA=3，OP=6，⊙O的圆心为O，圆O的切线为PA。求∠APB的大小。 | As shown in the diagram, OA=3, OP=6, O is the center of circle O, PA is the tangent to circle O. Find the measure of ∠APB. | 6777.png | [
"Shape(OA,AB,BO)",
"Shape(BA,OAB)",
"Shape(BA,AP)",
"Shape(AO,OB,OBA)",
"Shape(AP,PB,OAB)",
"Collinear(OBP)",
"Cocircular(O,AB)"
] | [
"Equal(LengthOfLine(OA),3)",
"Equal(LengthOfLine(OP),6)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PA,O)"
] | [] | Value(MeasureOfAngle(APB)) | 30 | [
"tangent_of_circle_property_perpendicular(1,PA,O,O)",
"right_triangle_judgment_angle(1,OAP)",
"right_triangle_property_pythagorean(1,OAP)",
"cosine_theorem(1,POA)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,PA,O,O)", "cosine_theorem(1,POA)"], "right_triangle_judgment_angle(1,OAP)": ["right_triangle_property_pythagorean(1,OAP)"], "tangent_of_circle_property_perpendicular(1,PA,O,O)": ["right_triangle_judgment_angle(1,OAP)"]} | |
6,778 | YangLi_2023-06-04 | GeoQA-9885 | 5 | 如图所示，OA=2，OC=2，∠CAO=30°，O是圆O的圆心，⊙O的切线为DC。求直线CD的长度。 | As shown in the diagram, OA=2, OC=2, ∠CAO=30°, O is the center of circle O, DC is the tangent to circle O. Find the length of line CD. | 6778.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OC)",
"Shape(CO,OB,OBC)",
"Shape(BD,DC,OBC)",
"Shape(BO,OA,OAB)",
"Collinear(AOBD)",
"Cocircular(O,ABC)"
] | [
"Equal(LengthOfLine(OA),2)",
"Equal(LengthOfLine(OC),2)",
"Equal(MeasureOfAngle(CAO),30)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(DC,O)"
] | [] | Value(LengthOfLine(CD)) | 2*sqrt(3) | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"triangle_property_angle_sum(1,COD)",
"tangent_of_circle_property_perpendicular(2,DC,O,O)",
"sine_theorem(1,COD)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "triangle_property_angle_sum(1,COD)", "tangent_of_circle_property_perpendicular(2,DC,O,O)", "sine_theorem(1,COD)"]} | |
6,779 | YangLi_2023-06-04 | GeoQA-9892 | 3 | 如图所示，AB=4，OA=5，圆O的圆心为O，⊙O的切线为AB。求直线OB的长度。 | As shown in the diagram, AB=4, OA=5, the center of ⊙O is O, AB is the tangent to circle O. Find the length of line OB. | 6779.png | [
"Shape(BO,OC,OCB)",
"Shape(CO,OB,OBC)",
"Shape(CA,AB,OCB)",
"Collinear(ACO)",
"Cocircular(O,CB)"
] | [
"Equal(LengthOfLine(AB),4)",
"Equal(LengthOfLine(OA),5)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AB,O)"
] | [] | Value(LengthOfLine(OB)) | 3 | [
"tangent_of_circle_property_perpendicular(2,AB,O,O)",
"right_triangle_judgment_angle(1,ABO)",
"right_triangle_property_pythagorean(1,ABO)"
] | {"START": ["tangent_of_circle_property_perpendicular(2,AB,O,O)"], "right_triangle_judgment_angle(1,ABO)": ["right_triangle_property_pythagorean(1,ABO)"], "tangent_of_circle_property_perpendicular(2,AB,O,O)": ["right_triangle_judgment_angle(1,ABO)"]} | |
6,780 | YangLi_2023-06-04 | GeoQA-9894 | 4 | 如图所示，AB=2，AC=1，∠BDA=90°，∠CAB=90°。求直线AD的长度。 | As shown in the diagram, AB=2, AC=1, ∠BDA=90°, ∠CAB=90°. Find the length of line AD. | 6780.png | [
"Shape(CA,AD,DC)",
"Shape(DA,AB,BD)",
"Collinear(CDB)"
] | [
"Equal(LengthOfLine(AB),2)",
"Equal(LengthOfLine(AC),1)",
"Equal(MeasureOfAngle(BDA),90)",
"Equal(MeasureOfAngle(CAB),90)"
] | [] | Value(LengthOfLine(AD)) | 2*sqrt(5)/5 | [
"right_triangle_judgment_angle(1,CAB)",
"right_triangle_property_pythagorean(1,CAB)",
"cosine_theorem(1,BCA)",
"sine_theorem(1,ABD)"
] | {"START": ["right_triangle_judgment_angle(1,CAB)", "cosine_theorem(1,BCA)", "sine_theorem(1,ABD)"], "right_triangle_judgment_angle(1,CAB)": ["right_triangle_property_pythagorean(1,CAB)"]} | |
6,781 | YangLi_2023-06-04 | GeoQA-9897 | 3 | 如图所示，OB=4，OC=4，∠BOC=60°，⊙O的圆心为O，AC是⊙O的切线。求直线AO的长度。 | As shown in the diagram, OB=4, OC=4, ∠BOC=60°, the center of ⊙O is O, AC is the tangent to ⊙O. Find the length of line AO. | 6781.png | [
"Shape(BO,OC,OCB)",
"Shape(CO,OB,OBC)",
"Shape(CA,AB,OCB)",
"Collinear(ABO)",
"Cocircular(O,BC)"
] | [
"Equal(LengthOfLine(OB),4)",
"Equal(LengthOfLine(OC),4)",
"Equal(MeasureOfAngle(BOC),60)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AC,O)"
] | [] | Value(LengthOfLine(AO)) | 8 | [
"tangent_of_circle_property_perpendicular(1,AC,O,O)",
"triangle_property_angle_sum(1,OCA)",
"sine_theorem(1,OCA)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,AC,O,O)", "triangle_property_angle_sum(1,OCA)", "sine_theorem(1,OCA)"]} | |
6,782 | YangLi_2023-06-04 | GeoQA-9901 | 4 | 如图所示，∠DAO=35°，O是圆O的圆心，⊙O的直径为AB，CD是圆O的切线。求∠BCD的大小。 | As shown in the diagram, ∠DAO=35°, O is the center of circle O, the diameter of circle O is AB, CD is the tangent to ⊙O. Find the measure of ∠BCD. | 6782.png | [
"Shape(AD,ODA)",
"Shape(DA,AO,OD)",
"Shape(DO,OB,OBD)",
"Shape(BO,OA,OAB)",
"Shape(BC,CD,OBD)",
"Collinear(AOBC)",
"Cocircular(O,ABD)"
] | [
"Equal(MeasureOfAngle(DAO),35)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(CD,O)"
] | [] | Value(MeasureOfAngle(BCD)) | 20 | [
"arc_property_center_angle(1,OBD,O)",
"arc_property_circumference_angle_external(1,OBD,A)",
"tangent_of_circle_property_perpendicular(2,CD,O,O)",
"triangle_property_angle_sum(1,DOC)"
] | {"START": ["arc_property_center_angle(1,OBD,O)", "arc_property_circumference_angle_external(1,OBD,A)", "tangent_of_circle_property_perpendicular(2,CD,O,O)", "triangle_property_angle_sum(1,DOC)"]} | |
6,783 | YangLi_2023-06-04 | GeoQA-9902 | 8 | 如图所示，∠BAP=40°，O是圆O的圆心，圆O的切线为AP。求∠APB的大小。 | As shown in the diagram, ∠BAP=40°, the center of circle O is O, the tangent to ⊙O is AP. Find the measure of ∠APB. | 6783.png | [
"Shape(BA,AP,OBP)",
"Shape(BP,PO,OB)",
"Shape(PB,OBP)",
"Shape(BO,OP,OPB)",
"Collinear(OBA)",
"Cocircular(O,BP)"
] | [
"Equal(MeasureOfAngle(BAP),40)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AP,O)"
] | [] | Value(MeasureOfAngle(APB)) | 25 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OP,O)",
"tangent_of_circle_property_perpendicular(2,AP,O,O)",
"triangle_property_angle_sum(1,POA)",
"isosceles_triangle_judgment_line_equal(1,OBP)",
"isosceles_triangle_property_angle_equal(1,OBP)",
"triangle_pro... | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OP,O)", "tangent_of_circle_property_perpendicular(2,AP,O,O)", "triangle_property_angle_sum(1,POA)", "triangle_property_angle_sum(1,POB)", "angle_addition(1,APB,BPO)"], "isosceles_triangle_judgment_line_equal(1,OBP)": ... | |
6,784 | YangLi_2023-06-04 | GeoQA-9903 | 3 | 如图所示，AB=4，AO=6，O是⊙O的圆心，⊙O的切线为AB。求直线OB的长度。 | As shown in the diagram, AB=4, AO=6, O is the center of circle O, AB is the tangent to circle O. Find the length of line OB. | 6784.png | [
"Shape(BO,OC,OCB)",
"Shape(CO,OB,OBC)",
"Shape(CA,AB,OCB)",
"Collinear(ACO)",
"Cocircular(O,CB)"
] | [
"Equal(LengthOfLine(AB),4)",
"Equal(LengthOfLine(AO),6)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AB,O)"
] | [] | Value(LengthOfLine(OB)) | 2*sqrt(5) | [
"tangent_of_circle_property_perpendicular(2,AB,O,O)",
"right_triangle_judgment_angle(1,ABO)",
"right_triangle_property_pythagorean(1,ABO)"
] | {"START": ["tangent_of_circle_property_perpendicular(2,AB,O,O)"], "right_triangle_judgment_angle(1,ABO)": ["right_triangle_property_pythagorean(1,ABO)"], "tangent_of_circle_property_perpendicular(2,AB,O,O)": ["right_triangle_judgment_angle(1,ABO)"]} | |
6,785 | YangLi_2023-06-04 | GeoQA-9905 | 6 | 如图所示，AB=8，AH=HB，OA=5，O是圆O的圆心，AH⊥OH。求直线HC的长度。 | As shown in the diagram, AB=8, AH=HB, OA=5, O is the center of ⊙O, AH is perpendicular to OH. Find the length of line HC. | 6785.png | [
"Shape(OA,AH,HO)",
"Shape(CH,HA,OAH)",
"Shape(BH,HC,OCB)",
"Shape(AO,OH,HB,OBA)",
"Collinear(OHC)",
"Collinear(AHB)",
"Cocircular(O,ACB)"
] | [
"Equal(LengthOfLine(AB),8)",
"Equal(LengthOfLine(AH),LengthOfLine(HB))",
"Equal(LengthOfLine(OA),5)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(AH,OH)"
] | [] | Value(LengthOfLine(HC)) | 2 | [
"line_addition(1,AH,HB)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"right_triangle_judgment_angle(1,AHO)",
"right_triangle_property_pythagorean(1,AHO)",
"line_addition(1,OH,HC)"
] | {"START": ["line_addition(1,AH,HB)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "right_triangle_judgment_angle(1,AHO)", "line_addition(1,OH,HC)"], "right_triangle_judgment_angle(1,AHO)": ["right_triangle_property_pythagorean(1,AHO)"]} | |
6,786 | YangLi_2023-06-04 | GeoQA-9907 | 4 | 如图所示，∠CAO=35°，O是圆O的圆心，⊙O的切线为PC。求∠BPC的大小。 | As shown in the diagram, ∠CAO=35°, the center of ⊙O is O, the tangent to ⊙O is PC. Find the measure of ∠BPC. | 6786.png | [
"Shape(AC,OCA)",
"Shape(CA,AO,OC)",
"Shape(CO,OB,OBC)",
"Shape(BP,PC,OBC)",
"Collinear(AOBP)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAO),35)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PC,O)"
] | [] | Value(MeasureOfAngle(BPC)) | 20 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"tangent_of_circle_property_perpendicular(2,PC,O,O)",
"triangle_property_angle_sum(1,COP)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "tangent_of_circle_property_perpendicular(2,PC,O,O)", "triangle_property_angle_sum(1,COP)"]} | |
6,787 | YangLi_2023-06-04 | GeoQA-9909 | 6 | 如图所示，PA=2，PO=5，O是圆O的圆心，圆O的切线为PB。求直线PB的长度。 | As shown in the diagram, PA=2, PO=5, the center of circle O is O, the tangent to ⊙O is PB. Find the length of line PB. | 6787.png | [
"Shape(AO,OB,OBA)",
"Shape(BP,PA,OBA)",
"Shape(BO,OA,OAB)",
"Collinear(PAO)",
"Cocircular(O,AB)"
] | [
"Equal(LengthOfLine(PA),2)",
"Equal(LengthOfLine(PO),5)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PB,O)"
] | [] | Value(LengthOfLine(PB)) | 4 | [
"line_addition(1,PA,AO)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"tangent_of_circle_property_perpendicular(1,PB,O,O)",
"right_triangle_judgment_angle(1,OBP)",
"right_triangle_property_pythagorean(1,OBP)"
] | {"START": ["line_addition(1,PA,AO)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "tangent_of_circle_property_perpendicular(1,PB,O,O)"], "right_triangle_judgment_angle(1,OBP)": ["right_triangle_property_pythagorean(1,OBP)"], "tangent_of_circle_property_perpendicula... | |
6,788 | YangLi_2023-06-04 | GeoQA-9910 | 7 | 如图所示，∠BOC=76°，O是⊙O的圆心，EC是圆O的切线。求∠BCE的大小。 | As shown in the diagram, ∠BOC=76°, the center of ⊙O is O, EC is the tangent to ⊙O. Find the measure of ∠BCE. | 6788.png | [
"Shape(AC,CO,OB,BA)",
"Shape(OC,CB,BO)",
"Shape(BC,OCB)",
"Shape(BC,CE)",
"Shape(AB,OBA)",
"Shape(CA,OAC)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(BOC),76)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(EC,O)"
] | [] | Value(MeasureOfAngle(BCE)) | 38 | [
"tangent_of_circle_property_perpendicular(1,EC,O,O)",
"angle_addition(1,OCB,BCE)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OCB)",
"isosceles_triangle_property_angle_equal(1,OCB)",
"triangle_property_ang... | {"START": ["tangent_of_circle_property_perpendicular(1,EC,O,O)", "angle_addition(1,OCB,BCE)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,OCB)"], "isosceles_triangle_judgment_line_equal(1,OCB)": ["isosceles_triangle_property_angle_eq... | |
6,789 | YangLi_2023-06-04 | GeoQA-9912 | 6 | 如图所示，AC=10，CO=10，OB=10，O是⊙O的圆心，圆O的切线为AE。求直线AE的长度。 | As shown in the diagram, AC=10, CO=10, OB=10, O is the center of circle O, AE is the tangent to ⊙O. Find the length of line AE. | 6789.png | [
"Shape(EA,AC,OEC)",
"Shape(CO,OE,OEC)",
"Shape(EO,OB,OBE)",
"Shape(BO,OC,OCB)",
"Collinear(ACOB)",
"Cocircular(O,ECB)"
] | [
"Equal(LengthOfLine(AC),10)",
"Equal(LengthOfLine(CO),10)",
"Equal(LengthOfLine(OB),10)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AE,O)"
] | [] | Value(LengthOfLine(AE)) | 10*sqrt(3) | [
"radius_of_circle_property_length_equal(1,OE,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"line_addition(1,AC,CO)",
"tangent_of_circle_property_perpendicular(1,AE,O,O)",
"right_triangle_judgment_angle(1,OEA)",
"right_triangle_property_pythagorean(1,OEA)"
] | {"START": ["radius_of_circle_property_length_equal(1,OE,O)", "radius_of_circle_property_length_equal(1,OB,O)", "line_addition(1,AC,CO)", "tangent_of_circle_property_perpendicular(1,AE,O,O)"], "right_triangle_judgment_angle(1,OEA)": ["right_triangle_property_pythagorean(1,OEA)"], "tangent_of_circle_property_perpendicula... | |
6,790 | YangLi_2023-06-04 | GeoQA-9913 | 9 | 如图所示，∠ABC=50°，O是圆O的圆心，圆O的切线为EA。求∠EAC的大小。 | As shown in the diagram, ∠ABC=50°, O is the center of ⊙O, the tangent to circle O is EA. Find the measure of ∠EAC. | 6790.png | [
"Shape(EA,AC)",
"Shape(AB,BC,CO,OA)",
"Shape(AO,OC,CA)",
"Shape(AC,OCA)",
"Shape(BA,OAB)",
"Shape(CB,OBC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABC),50)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(EA,O)"
] | [] | Value(MeasureOfAngle(EAC)) | 50 | [
"tangent_of_circle_property_perpendicular(2,EA,O,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"isosceles_triangle_judgment_line_equal(1,OCA)",
"isoscele... | {"START": ["tangent_of_circle_property_perpendicular(2,EA,O,O)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OCA,B)", "triangle_property_angle_sum(1,AOC)", "angle_addition(1,EAC,CAO... | |
6,791 | YangLi_2023-06-04 | GeoQA-9915 | 6 | 如图所示，∠ABO=65°，⊙O的圆心为O，CA是⊙O的切线。求∠BAC的大小。 | As shown in the diagram, ∠ABO=65°, O is the center of ⊙O, CA is the tangent to ⊙O. Find the measure of ∠BAC. | 6791.png | [
"Shape(AO,OB,OBA)",
"Shape(OA,AB,BO)",
"Shape(BA,OAB)",
"Shape(CA,OAB)",
"Cocircular(O,AB)"
] | [
"Equal(MeasureOfAngle(ABO),65)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(CA,O)"
] | [] | Value(MeasureOfAngle(BAC)) | 25 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"isosceles_triangle_property_angle_equal(1,OAB)",
"tangent_of_circle_property_perpendicular(1,CA,O,O)",
"angle_addition(1,OAB,BAC)"
] | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "tangent_of_circle_property_perpendicular(1,CA,O,O)", "angle_addition(1,OAB,BAC)"], "isosceles_triangle_judgment_line_equal(1,OAB)": ["isosceles_triangle_property_angle_equal(1,OAB)"], "radius_of_circle_proper... | |
6,792 | ChenyangMao_2023-06-04 | GeoQA-9916 | 4 | 如图所示，∠BCA=40°，O是⊙O的圆心，圆O的直径为BC，DA是⊙O的切线。求∠DAC的大小。 | As shown in the diagram, ∠BCA=40°, O is the center of ⊙O, the diameter of ⊙O is BC, the tangent to circle O is DA. Find the measure of ∠DAC. | 6792.png | [
"Shape(DA,AC)",
"Shape(OCA,AC)",
"Shape(OAB,BA)",
"Shape(BO,OC,CA,AB)",
"Shape(OBC,CO,OB)",
"Collinear(BOC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCA),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BC,O)",
"IsTangentOfCircle(DA,O)"
] | [] | Value(MeasureOfAngle(DAC)) | 50 | [
"diameter_of_circle_property_right_angle(1,CAB,O)",
"triangle_property_angle_sum(1,BCA)",
"arc_property_circumference_angle_external(1,OCA,B)",
"circle_property_angle_of_osculation(2,OCA,D)"
] | {"START": ["diameter_of_circle_property_right_angle(1,CAB,O)", "triangle_property_angle_sum(1,BCA)", "arc_property_circumference_angle_external(1,OCA,B)", "circle_property_angle_of_osculation(2,OCA,D)"]} | |
6,793 | ChenyangMao_2023-06-04 | GeoQA-9917 | 2 | 如图所示，∠APO=36°，圆O的圆心为O，圆O的切线为PA。求∠POA的大小。 | As shown in the diagram, ∠APO=36°, O is the center of circle O, the tangent to ⊙O is PA. Find the measure of ∠POA. | 6793.png | [
"Shape(OAB,BO,OA)",
"Shape(OBA,AO,OB)",
"Shape(AP,PB,OAB)",
"Collinear(OBP)",
"Cocircular(O,AB)"
] | [
"Equal(MeasureOfAngle(APO),36)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PA,O)"
] | [] | Value(MeasureOfAngle(POA)) | 54 | [
"tangent_of_circle_property_perpendicular(1,PA,O,O)",
"triangle_property_angle_sum(1,OAP)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,PA,O,O)", "triangle_property_angle_sum(1,OAP)"]} | |
6,794 | ChenyangMao_2023-06-04 | GeoQA-9918 | 6 | 如图所示，AC=4，BC=3，圆O的圆心为O，圆O的直径为AC，三角形ACB为直角三角形。求直线AD的长度。 | As shown in the diagram, AC=4, BC=3, the center of circle O is O, AC is the diameter of circle O, triangle ACB is a right triangle. Find the length of line AD. | 6794.png | [
"Shape(ODA,AD)",
"Shape(OAC,CO,OA)",
"Shape(AO,OC,CD,DA)",
"Shape(OCD,DC)",
"Shape(CB,BD,OCD)",
"Collinear(AOC)",
"Collinear(ADB)",
"Cocircular(O,ACD)"
] | [
"Equal(LengthOfLine(AC),4)",
"Equal(LengthOfLine(BC),3)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AC,O)",
"RightTriangle(ACB)"
] | [] | Value(LengthOfLine(AD)) | 16/5 | [
"diameter_of_circle_property_right_angle(1,CDA,O)",
"right_triangle_judgment_angle(1,CDA)",
"cosine_theorem(1,ACB)",
"cosine_theorem(1,ACD)",
"right_triangle_property_pythagorean(1,ACB)",
"right_triangle_property_pythagorean(1,CDA)"
] | {"START": ["diameter_of_circle_property_right_angle(1,CDA,O)", "cosine_theorem(1,ACB)", "cosine_theorem(1,ACD)", "right_triangle_property_pythagorean(1,ACB)"], "diameter_of_circle_property_right_angle(1,CDA,O)": ["right_triangle_judgment_angle(1,CDA)"], "right_triangle_judgment_angle(1,CDA)": ["right_triangle_property_... | |
6,795 | ChenyangMao_2023-06-04 | GeoQA-9922 | 6 | 如图所示，AB=2，圆O的半径为2，O是⊙O的圆心，EB是⊙O的直径，⊙O的切线为AD，AC⊥BC。求直线BC的长度。 | As shown in the diagram, AB=2, the radius of ⊙O is 2, the center of circle O is O, EB is the diameter of ⊙O, AD is the tangent to ⊙O, AC is perpendicular to BC. Find the length of line BC. | 6795.png | [
"Shape(ODE,EO,OD)",
"Shape(OBD,DO,OB)",
"Shape(BC,CD,OBD)",
"Shape(AC,CB,BA)",
"Collinear(EOBA)",
"Collinear(DCA)",
"Cocircular(O,BDE)"
] | [
"Equal(LengthOfLine(AB),2)",
"Equal(RadiusOfCircle(O),2)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(EB,O)",
"IsTangentOfCircle(AD,O)",
"PerpendicularBetweenLine(AC,BC)"
] | [] | Value(LengthOfLine(BC)) | 1 | [
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"tangent_of_circle_property_perpendicular(2,AD,O,O)",
"parallel_judgment_corresponding_angle(1,CB,DO,A)",
"midsegment_of_triangle_judgment_parallel(2,CB,ADO)",
"midsegment_of_triangle_property_length(1,CB,A... | {"START": ["radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OD,O)", "tangent_of_circle_property_perpendicular(2,AD,O,O)"], "midsegment_of_triangle_judgment_parallel(2,CB,ADO)": ["midsegment_of_triangle_property_length(1,CB,ADO)"], "parallel_judgment_corresponding_angle(1,CB,DO... | |
6,796 | ChenyangMao_2023-06-04 | GeoQA-angle-10002 | 4 | 如图所示，∠EGF=40°，三角形ABC为等边三角形，ED∥GF。求∠LDF的大小。 | As shown in the diagram, ∠EGF=40°, triangle ABC is an equilateral triangle, ED∥GF. Find the measure of ∠LDF. | 6796.png | [
"Shape(AE,ED,DA)",
"Shape(LD,DF)",
"Shape(DE,EG,GF,FD)",
"Shape(GB,BC,CF,FG)",
"Collinear(AEGB)",
"Collinear(EDL)",
"Collinear(ADFC)"
] | [
"Equal(MeasureOfAngle(EGF),40)",
"EquilateralTriangle(ABC)",
"ParallelBetweenLine(ED,GF)"
] | [] | Value(MeasureOfAngle(LDF)) | 80 | [
"equilateral_triangle_property_angle(1,ABC)",
"parallel_property_corresponding_angle(1,ED,GF,A)",
"triangle_property_angle_sum(1,AED)",
"vertical_angle(1,EDA,LDF)"
] | {"START": ["equilateral_triangle_property_angle(1,ABC)", "parallel_property_corresponding_angle(1,ED,GF,A)", "triangle_property_angle_sum(1,AED)", "vertical_angle(1,EDA,LDF)"]} | |
6,797 | ChenyangMao_2023-06-04 | GeoQA-angle-10003 | 4 | 如图所示，∠DAB=20°，BD是CA的垂直平分线。求∠DBC的大小。 | As shown in the diagram, ∠DAB=20°, BD perpendicular bisects CA. Find the measure of ∠DBC. | 6797.png | [
"Shape(DA,AB,BD)",
"Shape(CD,DB,BC)",
"Collinear(ADC)"
] | [
"Equal(MeasureOfAngle(DAB),20)",
"IsPerpendicularBisectorOfLine(BD,CA)"
] | [] | Value(MeasureOfAngle(DBC)) | 70 | [
"perpendicular_bisector_property_distance_equal(1,BD,CA)",
"isosceles_triangle_judgment_line_equal(1,BCA)",
"isosceles_triangle_property_angle_equal(1,BCA)",
"triangle_property_angle_sum(1,CDB)"
] | {"START": ["perpendicular_bisector_property_distance_equal(1,BD,CA)", "triangle_property_angle_sum(1,CDB)"], "isosceles_triangle_judgment_line_equal(1,BCA)": ["isosceles_triangle_property_angle_equal(1,BCA)"], "perpendicular_bisector_property_distance_equal(1,BD,CA)": ["isosceles_triangle_judgment_line_equal(1,BCA)"]} | |
6,798 | ChenyangMao_2023-06-04 | GeoQA-angle-10004 | 9 | 如图所示，∠ABC=20°，∠BCA=30°，PM垂直平分AB，QN是CA的垂直平分线。求∠QAP的大小。 | As shown in the diagram, ∠ABC=20°, ∠BCA=30°, PM is the perpendicular bisector of AB, QN perpendicular bisects CA. Find the measure of ∠QAP. | 6798.png | [
"Shape(AM,MP,PA)",
"Shape(MB,BP,PM)",
"Shape(AP,PQ,QA)",
"Shape(NA,AQ,QN)",
"Shape(CN,NQ,QC)",
"Collinear(ANC)",
"Collinear(AMB)",
"Collinear(BPQC)"
] | [
"Equal(MeasureOfAngle(ABC),20)",
"Equal(MeasureOfAngle(BCA),30)",
"IsPerpendicularBisectorOfLine(PM,AB)",
"IsPerpendicularBisectorOfLine(QN,CA)"
] | [] | Value(MeasureOfAngle(QAP)) | 80 | [
"perpendicular_bisector_property_distance_equal(1,PM,AB)",
"perpendicular_bisector_property_distance_equal(1,QN,CA)",
"isosceles_triangle_judgment_line_equal(1,PAB)",
"isosceles_triangle_judgment_line_equal(1,QCA)",
"isosceles_triangle_property_angle_equal(1,PAB)",
"isosceles_triangle_property_angle_equal... | {"START": ["perpendicular_bisector_property_distance_equal(1,PM,AB)", "perpendicular_bisector_property_distance_equal(1,QN,CA)", "angle_addition(1,NAQ,QAP)", "angle_addition(1,NAP,PAM)", "triangle_property_angle_sum(1,BCA)"], "isosceles_triangle_judgment_line_equal(1,PAB)": ["isosceles_triangle_property_angle_equal(1,P... | |
6,799 | ChenyangMao_2023-06-04 | GeoQA-angle-10005 | 7 | 如图所示，∠DAC=52°，圆B的圆心为B，AC和CB是直角三角形ACB的两个直角边。求∠CDA的大小。 | As shown in the diagram, ∠DAC=52°, B is the center of ⊙B, △ACB is a right △. Find the measure of ∠CDA. | 6799.png | [
"Shape(DA,AC,BDC)",
"Shape(BDC,CD)",
"Shape(BD,DC,CB)",
"Collinear(ADB)",
"Cocircular(B,DC)"
] | [
"Equal(MeasureOfAngle(DAC),52)",
"IsCentreOfCircle(B,B)",
"RightTriangle(ACB)"
] | [] | Value(MeasureOfAngle(CDA)) | 109 | [
"triangle_property_angle_sum(1,BDC)",
"triangle_property_angle_sum(1,ACB)",
"radius_of_circle_property_length_equal(1,BD,B)",
"radius_of_circle_property_length_equal(1,BC,B)",
"isosceles_triangle_judgment_line_equal(1,BDC)",
"isosceles_triangle_property_angle_equal(1,BDC)",
"adjacent_complementary_angle... | {"START": ["triangle_property_angle_sum(1,BDC)", "triangle_property_angle_sum(1,ACB)", "radius_of_circle_property_length_equal(1,BD,B)", "radius_of_circle_property_length_equal(1,BC,B)", "adjacent_complementary_angle(1,BDC,CDA)"], "isosceles_triangle_judgment_line_equal(1,BDC)": ["isosceles_triangle_property_angle_equa... | |
6,800 | ChenyangMao_2023-06-04 | GeoQA-angle-10006 | 10 | 如图所示，AB=AC，∠CAB=100°，CE是∠BCA的角平分线，AD是△ABC的中线。求∠AFC的大小。 | As shown in the diagram, AB=AC, ∠CAB=100°, CE bisects ∠BCA, AD is the median of triangle ABC. Find the measure of ∠AFC. | 6800.png | [
"Shape(AE,EF,FA)",
"Shape(EB,BD,DF,FE)",
"Shape(AF,FC,CA)",
"Shape(FD,DC,CF)",
"Collinear(AEB)",
"Collinear(AFD)",
"Collinear(EFC)",
"Collinear(BDC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(CAB),100)",
"IsBisectorOfAngle(CE,BCA))",
"IsMedianOfTriangle(AD,ABC)"
] | [] | Value(MeasureOfAngle(AFC)) | 110 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"triangle_property_angle_sum(1,AFC)",
"triangle_property_angle_sum(1,FDC)",
"triangle_property_angle_sum(1,CAD)",
"triangle_property_angle_sum(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"mirror_congruent_triangle_judgment_sas(1,BDA,CAD)",
... | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,AFC)", "triangle_property_angle_sum(1,FDC)", "triangle_property_angle_sum(1,CAD)", "triangle_property_angle_sum(1,ABC)", "adjacent_complementary_angle(1,BDA,FDC)", "adjacent_complementary_angle(1,AFC,CFD)"], "isosceles_triangle_j... |
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