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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6,901 | ZheZhu_2023-06-04 | GeoQA-angle-10142 | 8 | 如图所示，AB=OA，⊙O的圆心为O。求∠BDA的大小。 | As shown in the diagram, AB=OA, the center of ⊙O is O. Find the measure of ∠BDA. | 6901.png | [
"Shape(BO,OE,EB)",
"Shape(BE,EA,AB)",
"Shape(OAB,BA)",
"Shape(ED,DA,AE)",
"Shape(ODA,AD)",
"Collinear(OEA)",
"Collinear(BED)",
"Cocircular(O,ABCD)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(OA))",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BDA)) | 30 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,BOA)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"equilateral_triangle_judgment_isosceles_and_isosceles(1,BOA)",
"equilateral_triangle_property_angle(1,OAB)"... | {"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,D)"], "equilateral_triangle_judgment_isosceles_and_isosceles(1,BOA)": ["equilateral_triangle_property_angle(1,OAB)"], "iso... | |
6,902 | ZheZhu_2023-06-04 | GeoQA-angle-10144 | 5 | 如图所示，∠BOC=130°，圆O的圆心为O，AB是⊙O的直径。求∠CDA的大小。 | As shown in the diagram, ∠BOC=130°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠CDA. | 6902.png | [
"Shape(OC,OCB,BO)",
"Shape(CO,OE,EC)",
"Shape(ED,DA,AE)",
"Collinear(AEOB)",
"Collinear(CED)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(BOC),130)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDA)) | 25 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_center_angle(1,OCB,O)",
"arc_property_circumference_angle_external(1,OAC,D)",
"arc_addition_measure(1,OAC,OCB)",
"flat_angle(1,BOA)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,OAC,D)", "arc_addition_measure(1,OAC,OCB)", "flat_angle(1,BOA)"]} | |
6,903 | ZheZhu_2023-06-04 | GeoQA-angle-10145 | 2 | 如图所示，∠AOB=122°，O是圆O的圆心。求∠BCA的大小。 | As shown in the diagram, ∠AOB=122°, O is the center of circle O. Find the measure of ∠BCA. | 6903.png | [
"Shape(CA,AO,OB,BC)",
"Shape(OA,AD,DB,BO)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(AOB),122)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 119 | [
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_angle_internal(1,OBA,C)"
] | {"START": ["arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_internal(1,OBA,C)"]} | |
6,904 | ZheZhu_2023-06-04 | GeoQA-angle-10149 | 5 | 如图所示，Add(MeasureOfAngle(ADC)=MeasureOfAngle(CBA))，∠BQA=110°，⊙O的圆心为O，圆O的切线为QA，⊙O的切线为QB。求∠ADC的大小与∠CBQ的大小之和。 | As shown in the diagram, Add(MeasureOfAngle(ADC)=MeasureOfAngle(CBA)), ∠BQA=110°, the center of ⊙O is O, QA is the tangent to ⊙O, QB is the tangent to circle O. Find the sum of the measure of ∠ADC and the measure of ∠CBQ. | 6904.png | [
"Shape(AD,DC,CB,BA)",
"Shape(OAD,DA)",
"Shape(ODC,CD)",
"Shape(OCB,BC)",
"Shape(OBA,AB)",
"Shape(QA,OBA,BQ)",
"Cocircular(O,DCBA)"
] | [
"Equal(Add(MeasureOfAngle(ADC),MeasureOfAngle(CBA)),180)",
"Equal(MeasureOfAngle(BQA),110)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(QA,O)",
"IsTangentOfCircle(QB,O)"
] | [] | Value(Add(MeasureOfAngle(ADC),MeasureOfAngle(CBQ))) | 215 | [
"tangent_of_circle_property_length_equal(1,QA,QB,O)",
"isosceles_triangle_judgment_line_equal(1,QAB)",
"isosceles_triangle_property_angle_equal(1,QAB)",
"triangle_property_angle_sum(1,ABQ)",
"angle_addition(1,CBA,ABQ)"
] | {"START": ["tangent_of_circle_property_length_equal(1,QA,QB,O)", "triangle_property_angle_sum(1,ABQ)", "angle_addition(1,CBA,ABQ)"], "isosceles_triangle_judgment_line_equal(1,QAB)": ["isosceles_triangle_property_angle_equal(1,QAB)"], "tangent_of_circle_property_length_equal(1,QA,QB,O)": ["isosceles_triangle_judgment_li... | |
6,905 | ZheZhu_2023-06-04 | GeoQA-angle-10150 | 7 | 如图所示，CD=AD，DA=DC，∠BCD=90°，∠DCO=90°，∠OAB=90°，CF平分∠BCD，O是⊙O的圆心，AC是三角形ABO的中线。求∠EOC的大小。 | As shown in the diagram, CD=AD, DA=DC, ∠BCD=90°, ∠DCO=90°, ∠OAB=90°, CF is the angle bisector of ∠BCD, the center of ⊙O is O, AC is the median of △ ABO. Find the measure of ∠EOC. | 6905.png | [
"Shape(OC,CE,EO)",
"Shape(OA,AC,CO)",
"Shape(OAC,CA)",
"Shape(OAC,AD,DC)",
"Shape(CD,DF,FC)",
"Shape(CF,FB,BC)",
"Collinear(ECF)",
"Collinear(OCB)",
"Collinear(ADFB)",
"Cocircular(O,ACE)"
] | [
"Equal(LengthOfLine(CD),LengthOfLine(AD))",
"Equal(LengthOfLine(DA),LengthOfLine(DC))",
"Equal(MeasureOfAngle(BCD),90)",
"Equal(MeasureOfAngle(DCO),90)",
"Equal(MeasureOfAngle(OAB),90)",
"IsBisectorOfAngle(CF,BCD)",
"IsCentreOfCircle(O,O)",
"IsMedianOfTriangle(AC,ABO)"
] | [] | Value(MeasureOfAngle(EOC)) | 90 | [
"angle_addition(1,BCF,FCD)",
"vertical_angle(1,OCE,BCF)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OE,O)",
"isosceles_triangle_judgment_line_equal(1,OCE)",
"isosceles_triangle_property_angle_equal(1,OCE)",
"triangle_property_angle_sum(1,OCE)"
] | {"START": ["angle_addition(1,BCF,FCD)", "vertical_angle(1,OCE,BCF)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OE,O)", "triangle_property_angle_sum(1,OCE)"], "isosceles_triangle_judgment_line_equal(1,OCE)": ["isosceles_triangle_property_angle_equal(1,OCE)"], "radius_of_... | |
6,906 | ZheZhu_2023-06-04 | GeoQA-angle-10153 | 5 | 如图所示，∠CAB=35°，O是圆O的圆心，⊙O的直径为AB。求∠ADC的大小。 | As shown in the diagram, ∠CAB=35°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ADC. | 6906.png | [
"Shape(CA,AO,OE,EC)",
"Shape(CE,EB,BC)",
"Shape(OBC,CB)",
"Shape(EO,OA,AD,DE)",
"Shape(BE,ED,ODB)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(CAB),35)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ADC)) | 55 | [
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_angle_external(1,OCA,D)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_addition_measure(1,OBC,OCA)",
"flat_angle(1,AOE)"
] | {"START": ["arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_external(1,OCA,D)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_addition_measure(1,OBC,OCA)", "flat_angle(1,AOE)"]} | |
6,907 | ZheZhu_2023-06-04 | GeoQA-angle-10155 | 3 | 如图所示，⌒OAC的长度与弧OCB的长度相等，∠BOA=90°，⌒OAC的角度与⌒OCB的角度相等，O是圆O的圆心。求∠CPA的大小。 | As shown in the diagram, the length of ⌒OAC is equal to the length of arc OCB, ∠BOA=90°, the measure of arc OAC is equal to the measure of ⌒OCB, O is the center of ⊙O. Find the measure of ∠CPA. | 6907.png | [
"Shape(OA,AD,DO)",
"Shape(PD,DE,EP)",
"Shape(ED,DA,OAC,CE)",
"Shape(EC,OCB,BE)",
"Shape(PE,EB,OBP)",
"Collinear(ODEB)",
"Collinear(PDA)",
"Collinear(PEC)",
"Cocircular(O,ACBP)"
] | [
"Equal(LengthOfArc(OAC),LengthOfArc(OCB))",
"Equal(MeasureOfAngle(BOA),90)",
"Equal(MeasureOfArc(OAC),MeasureOfArc(OCB))",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CPA)) | 45/2 | [
"arc_addition_measure(1,OAC,OCB)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAC,P)"
] | {"START": ["arc_addition_measure(1,OAC,OCB)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAC,P)"]} | |
6,908 | ZheZhu_2023-06-04 | GeoQA-angle-10157 | 5 | 如图所示，∠CEA=64°，∠COA=90°，O是圆O的圆心，圆O的直径为AB。求∠DAB的大小。 | As shown in the diagram, ∠CEA=64°, ∠COA=90°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DAB. | 6908.png | [
"Shape(DA,AO,OE,ED)",
"Shape(OA,OAC,CO)",
"Shape(EO,OC,CE)",
"Shape(DE,EB,OBD)",
"Collinear(AOEB)",
"Collinear(DEC)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(CEA),64)",
"Equal(MeasureOfAngle(COA),90)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DAB)) | 19 | [
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,D)",
"triangle_property_angle_sum(1,DAE)",
"flat_angle(1,CED)",
"angle_addition(1,CEA,AED)"
] | {"START": ["arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,D)", "triangle_property_angle_sum(1,DAE)", "flat_angle(1,CED)", "angle_addition(1,CEA,AED)"]} | |
6,909 | ZheZhu_2023-06-04 | GeoQA-angle-10158 | 6 | 如图所示，∠COB=100°，⊙O的圆心为O，AD平行于CB，DA平行于BC。求∠ADC的大小。 | As shown in the diagram, ∠COB=100°, the center of ⊙O is O, AD is parallel to CB, DA is parallel to BC. Find the measure of ∠ADC. | 6909.png | [
"Shape(AD,DO,OC,CA)",
"Shape(OB,BC,CO)",
"Collinear(DOC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(COB),100)",
"IsCentreOfCircle(O,O)",
"ParallelBetweenLine(AD,CB)",
"ParallelBetweenLine(DA,BC)"
] | [] | Value(MeasureOfAngle(ADC)) | 40 | [
"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)",
"parallel_property_alternate_interior_angle(1,DA,BC)"
] | {"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)", "parallel_property_alternate_interior_angle(1,DA,BC)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property_angle_equal(1,OBC)"], "radius_of_cir... | |
6,910 | ZheZhu_2023-06-04 | GeoQA-angle-10159 | 5 | 如图所示，∠BCD=25°，∠DEA=39°，⊙O的圆心为O。求∠CDA的大小。 | As shown in the diagram, ∠BCD=25°, ∠DEA=39°, the center of circle O is O. Find the measure of ∠CDA. | 6910.png | [
"Shape(AF,FB,BA)",
"Shape(OCD,DC)",
"Shape(OBA,AB)",
"Shape(BF,FD,ODB)",
"Shape(FC,CD,DF)",
"Shape(ODB,DE,EB)",
"Collinear(ABE)",
"Collinear(CDE)",
"Collinear(CFB)",
"Collinear(AFD)",
"Cocircular(O,CDBA)"
] | [
"Equal(MeasureOfAngle(BCD),25)",
"Equal(MeasureOfAngle(DEA),39)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CDA)) | 64 | [
"arc_property_circumference_angle_external(1,ODB,C)",
"arc_property_circumference_angle_external(1,ODB,A)",
"triangle_property_angle_sum(1,ADE)",
"flat_angle(1,CDE)",
"angle_addition(1,CDF,FDE)"
] | {"START": ["arc_property_circumference_angle_external(1,ODB,C)", "arc_property_circumference_angle_external(1,ODB,A)", "triangle_property_angle_sum(1,ADE)", "flat_angle(1,CDE)", "angle_addition(1,CDF,FDE)"]} | |
6,911 | ZheZhu_2023-06-04 | GeoQA-angle-10161 | 10 | 如图所示，AB=AD，∠DOC=120°，圆O的圆心为O，圆O的直径为DB。求∠BGA的大小。 | As shown in the diagram, AB=AD, ∠DOC=120°, the center of ⊙O is O, DB is the diameter of circle O. Find the measure of ∠BGA. | 6911.png | [
"Shape(DA,AG,GO,OD)",
"Shape(CO,OG,GC)",
"Shape(GA,AB,BG)",
"Shape(ODA,AD)",
"Shape(DO,OC,OCD)",
"Collinear(DOGB)",
"Collinear(AGC)",
"Cocircular(O,DABC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AD))",
"Equal(MeasureOfAngle(DOC),120)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(DB,O)"
] | [] | Value(MeasureOfAngle(BGA)) | 105 | [
"flat_angle(1,DOG)",
"angle_addition(1,DOC,COG)",
"arc_property_center_angle(1,OBD,O)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,OBC,A)",
"isosceles_triangle_judgment_line_equal(1,ABD)",
"isosceles_triangle... | {"START": ["flat_angle(1,DOG)", "angle_addition(1,DOC,COG)", "arc_property_center_angle(1,OBD,O)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,OBC,A)", "isosceles_triangle_judgment_line_equal(1,ABD)", "triangle_property_angle_s... | |
6,912 | ZheZhu_2023-06-04 | GeoQA-angle-10162 | 4 | 如图所示，∠APO=45°，O是⊙O的圆心，圆O的切线为PA。求∠BCA的大小。 | As shown in the diagram, ∠APO=45°, O is the center of ⊙O, PA is the tangent to circle O. Find the measure of ∠BCA. | 6912.png | [
"Shape(OD,DB,BO)",
"Shape(BD,DA,OAB)",
"Shape(OAB,AP,PB)",
"Shape(DC,CA,AD)",
"Collinear(CDB)",
"Collinear(ODA)",
"Collinear(OBP)",
"Cocircular(O,CAB)"
] | [
"Equal(MeasureOfAngle(APO),45)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PA,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 45/2 | [
"tangent_of_circle_property_perpendicular(1,PA,O,O)",
"triangle_property_angle_sum(1,OAP)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,PA,O,O)", "triangle_property_angle_sum(1,OAP)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)"]} | |
6,913 | ZheZhu_2023-06-04 | GeoQA-angle-10164 | 2 | 如图所示，∠ABC=60°，圆O的圆心为O。求∠AOC的大小。 | As shown in the diagram, ∠ABC=60°, O is the center of circle O. Find the measure of ∠AOC. | 6913.png | [
"Shape(AO,OC,CA)",
"Shape(AB,BC,CO,OA)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABC),60)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOC)) | 120 | [
"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)"]} | |
6,914 | ZheZhu_2023-06-04 | GeoQA-angle-10165 | 7 | 如图所示，∠OBA=25°，O是圆O的圆心。求∠ACB的大小。 | As shown in the diagram, ∠OBA=25°, O is the center of ⊙O. Find the measure of ∠ACB. | 6914.png | [
"Shape(AO,OB,BA)",
"Shape(AC,CB,BO,OA)",
"Shape(OAC,CA)",
"Shape(OCB,BC)",
"Shape(OBA,AB)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(OBA),25)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ACB)) | 65 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OBA)",
"isosceles_triangle_property_angle_equal(1,OBA)",
"triangle_property_angle_sum(1,AOB)",
"arc_property_center_angle(1,OBA,O)",
"arc_property_circumference_a... | {"START": ["radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,AOB)", "arc_property_center_angle(1,OBA,O)", "arc_property_circumference_angle_external(1,OBA,C)"], "isosceles_triangle_judgment_line_equal(1,OBA)": ["isosceles_triangle_property... | |
6,915 | ZheZhu_2023-06-04 | GeoQA-angle-10167 | 4 | 如图所示，∠BCA=50°，⊙O的圆心为O，AD是⊙O的直径。求∠DAB的大小。 | As shown in the diagram, ∠BCA=50°, O is the center of circle O, the diameter of ⊙O is AD. Find the measure of ∠DAB. | 6915.png | [
"Shape(CA,AE,EC)",
"Shape(EA,AB,BE)",
"Shape(EB,BD,DO,OE)",
"Collinear(AEOD)",
"Collinear(BEC)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(BCA),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AD,O)"
] | [] | Value(MeasureOfAngle(DAB)) | 40 | [
"diameter_of_circle_property_right_angle(1,ABD,O)",
"arc_property_circumference_angle_external(1,OAB,D)",
"arc_property_circumference_angle_external(1,OAB,C)",
"triangle_property_angle_sum(1,ABD)"
] | {"START": ["diameter_of_circle_property_right_angle(1,ABD,O)", "arc_property_circumference_angle_external(1,OAB,D)", "arc_property_circumference_angle_external(1,OAB,C)", "triangle_property_angle_sum(1,ABD)"]} | |
6,916 | ZheZhu_2023-06-04 | GeoQA-angle-10168 | 6 | 如图所示，∠CAD=50°，O是圆O的圆心，圆O的直径为AD。求∠ABC的大小。 | As shown in the diagram, ∠CAD=50°, the center of circle O is O, the diameter of ⊙O is AD. Find the measure of ∠ABC. | 6916.png | [
"Shape(ED,DC,CE)",
"Shape(OE,EC,CA,AO)",
"Shape(BE,EO,OA,AB)",
"Collinear(BEC)",
"Collinear(DEOA)",
"Cocircular(O,BDCA)"
] | [
"Equal(MeasureOfAngle(CAD),50)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AD,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 40 | [
"arc_property_center_angle(1,OAD,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_circumference_angle_external(1,OCA,D)",
"arc_property_circumference_angle_external(1,OAD,C)",
"triangle_property_angle_sum(1,DCA)",
"flat_angle(1,DOA)"
] | {"START": ["arc_property_center_angle(1,OAD,O)", "arc_property_circumference_angle_external(1,OCA,B)", "arc_property_circumference_angle_external(1,OCA,D)", "arc_property_circumference_angle_external(1,OAD,C)", "triangle_property_angle_sum(1,DCA)", "flat_angle(1,DOA)"]} | |
6,917 | ZheZhu_2023-06-04 | GeoQA-angle-10169 | 17 | 如图所示，∠BEO=90°，∠CDB=70°，O是⊙O的圆心。求∠OAC的大小。 | As shown in the diagram, ∠BEO=90°, ∠CDB=70°, O is the center of circle O. Find the measure of ∠OAC. | 6917.png | [
"Shape(DB,BO,OC,CD)",
"Shape(OB,BE,EO)",
"Shape(OE,EC,CO)",
"Shape(CE,EA,AC)",
"Collinear(BEC)",
"Collinear(OEA)",
"Cocircular(O,DBAC)"
] | [
"Equal(MeasureOfAngle(BEO),90)",
"Equal(MeasureOfAngle(CDB),70)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OAC)) | 55 | [
"arc_property_center_angle(1,OBA,O)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OBC,D)",
"circle_property_chord_perpendicular_bisect_arc(1,OBC,OEA)",
"congruent_arc_judgment_length_equal(1,OBA,OAC)",
"congruent_arc_property_m... | {"START": ["arc_property_center_angle(1,OBA,O)", "arc_property_center_angle(1,OBC,O)", "arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OBC,D)", "circle_property_chord_perpendicular_bisect_arc(1,OBC,OEA)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_len... | |
6,918 | ZheZhu_2023-06-04 | GeoQA-angle-10170 | 4 | 如图所示，∠ABD=54°，O是圆O的圆心，AB是⊙O的直径。求∠DCB的大小。 | As shown in the diagram, ∠ABD=54°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DCB. | 6918.png | [
"Shape(AE,ED,DA)",
"Shape(DE,EO,OB,BD)",
"Shape(BO,OE,EC,CB)",
"Collinear(AEOB)",
"Collinear(DEC)",
"Cocircular(O,DACB)"
] | [
"Equal(MeasureOfAngle(ABD),54)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DCB)) | 36 | [
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,OBD,C)",
"diameter_of_circle_property_right_angle(1,BDA,O)",
"triangle_property_angle_sum(1,DAB)"
] | {"START": ["arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,OBD,C)", "diameter_of_circle_property_right_angle(1,BDA,O)", "triangle_property_angle_sum(1,DAB)"]} | |
6,919 | ZheZhu_2023-06-04 | GeoQA-angle-10171 | 2 | 如图所示，∠ABO=40°，∠COA=120°，∠OAB=90°。求∠COB的大小。 | As shown in the diagram, ∠ABO=40°, ∠COA=120°, ∠OAB=90°. Find the measure of ∠COB. | 6919.png | [
"Shape(AB,BO,OA)",
"Shape(CO,OB)"
] | [
"Equal(MeasureOfAngle(ABO),40)",
"Equal(MeasureOfAngle(COA),120)",
"Equal(MeasureOfAngle(OAB),90)"
] | [] | Value(MeasureOfAngle(COB)) | 70 | [
"triangle_property_angle_sum(1,ABO)",
"angle_addition(1,COB,BOA)"
] | {"START": ["triangle_property_angle_sum(1,ABO)", "angle_addition(1,COB,BOA)"]} | |
6,920 | ZheZhu_2023-06-04 | GeoQA-angle-10172 | 7 | 如图所示，∠OBC=35°，O是圆O的圆心。求∠CAB的大小。 | As shown in the diagram, ∠OBC=35°, the center of circle O is O. Find the measure of ∠CAB. | 6920.png | [
"Shape(AB,BO,OC,CA)",
"Shape(OB,BC,CO)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(OBC),35)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 55 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"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_pro... | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OBC)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property... | |
6,921 | ZheZhu_2023-06-04 | GeoQA-angle-10173 | 7 | 如图所示，∠OBC=25°，圆O的圆心为O。求∠CAB的大小。 | As shown in the diagram, ∠OBC=25°, O is the center of ⊙O. Find the measure of ∠CAB. | 6921.png | [
"Shape(AB,BO,OC,CA)",
"Shape(OB,BC,CO)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(OBC),25)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(CAB)) | 65 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"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_pro... | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OBC)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property... | |
6,922 | ZheZhu_2023-06-04 | GeoQA-angle-10174 | 6 | 如图所示，OA=2，∠BCA=45°，⊙O的圆心为O。求直线AB的长度。 | As shown in the diagram, OA=2, ∠BCA=45°, the center of circle O is O. Find the length of line AB. | 6922.png | [
"Shape(CA,AO,OB,BC)",
"Shape(OA,AB,BO)",
"Cocircular(O,CAB)"
] | [
"Equal(LengthOfLine(OA),2)",
"Equal(MeasureOfAngle(BCA),45)",
"IsCentreOfCircle(O,O)"
] | [] | Value(LengthOfLine(AB)) | 2*sqrt(2) | [
"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)",
"right_triangle_judgment_angle(1,BOA)",
"right_triangle_property_pythagorean(1,BOA)"
] | {"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)"], "arc_property_center_angle(1,OAB,O)": ["right_triangle_judgment_angle(1,BOA)"], "arc_property_circumference_angle_ex... | |
6,923 | ZheZhu_2023-06-04 | GeoQA-angle-10175 | 7 | 如图所示，∠CAB=74°，O是⊙O的圆心。求∠OBC的大小。 | As shown in the diagram, ∠CAB=74°, O is the center of circle O. Find the measure of ∠OBC. | 6923.png | [
"Shape(AB,BO,OC,CA)",
"Shape(OB,BC,CO)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),74)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OBC)) | 16 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"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_pro... | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,OBC)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property... | |
6,924 | ChengQin_2023-06-04 | GeoQA-angle-10176 | 4 | 如图所示，∠DBA=20°，∠DEA=80°，O是圆O的圆心。求∠COB的大小。 | As shown in the diagram, ∠DBA=20°, ∠DEA=80°, the center of ⊙O is O. Find the measure of ∠COB. | 6924.png | [
"Shape(OCA,AE,EC)",
"Shape(CE,EO,OC)",
"Shape(OAD,DE,EA)",
"Shape(ED,DB,BO,OE)",
"Shape(ODB,BD)",
"Shape(OBC,CO,OB)",
"Collinear(AEOB)",
"Collinear(DEC)",
"Cocircular(O,CADB)"
] | [
"Equal(MeasureOfAngle(DBA),20)",
"Equal(MeasureOfAngle(DEA),80)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(COB)) | 120 | [
"adjacent_complementary_angle(1,BED,DEA)",
"triangle_property_angle_sum(1,DBE)",
"arc_property_circumference_angle_external(1,OBC,D)",
"arc_property_center_angle(1,OBC,O)"
] | {"START": ["adjacent_complementary_angle(1,BED,DEA)", "triangle_property_angle_sum(1,DBE)", "arc_property_circumference_angle_external(1,OBC,D)", "arc_property_center_angle(1,OBC,O)"]} | |
6,925 | ChengQin_2023-06-04 | GeoQA-angle-10177 | 2 | 如图所示，∠BOA=70°，O是⊙O的圆心。求∠BCA的大小。 | As shown in the diagram, ∠BOA=70°, O is the center of circle O. Find the measure of ∠BCA. | 6925.png | [
"Shape(OD,DC,OCA,AO)",
"Shape(OA,AD,DO)",
"Shape(DA,OAB,BD)",
"Shape(DB,BC,CD)",
"Shape(OBC,CB)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BOA),70)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 35 | [
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_center_angle(1,OAB,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OAB,C)", "arc_property_center_angle(1,OAB,O)"]} | |
6,926 | ChengQin_2023-06-04 | GeoQA-angle-10178 | 1 | 如图所示，O是⊙O的圆心，⊙O的直径为AB。求∠BCA的大小。 | As shown in the diagram, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BCA. | 6926.png | [
"Shape(OB,BC,CA,AO)",
"Shape(OBC,CB)",
"Shape(OCA,AC)",
"Shape(OA,OAB,BO)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 90 | [
"diameter_of_circle_property_right_angle(1,BCA,O)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)"]} | |
6,927 | ChengQin_2023-06-04 | GeoQA-angle-10179 | 4 | 如图所示，∠ADC=25°，O是圆O的圆心，BA是⊙O的切线。求∠OBA的大小。 | As shown in the diagram, ∠ADC=25°, the center of circle O is O, BA is the tangent to circle O. Find the measure of ∠OBA. | 6927.png | [
"Shape(OE,ED,ODC,CO)",
"Shape(OC,CE,EO)",
"Shape(EA,AD,DE)",
"Shape(EC,OCA,AE)",
"Shape(OAD,DA)",
"Shape(CB,BA,OCA)",
"Collinear(BCO)",
"Collinear(CED)",
"Collinear(OEA)",
"Cocircular(O,CAD)"
] | [
"Equal(MeasureOfAngle(ADC),25)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(BA,O)"
] | [] | Value(MeasureOfAngle(OBA)) | 40 | [
"arc_property_circumference_angle_external(1,OCA,D)",
"arc_property_center_angle(1,OCA,O)",
"tangent_of_circle_property_perpendicular(2,BA,O,O)",
"triangle_property_angle_sum(1,OBA)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,D)", "arc_property_center_angle(1,OCA,O)", "tangent_of_circle_property_perpendicular(2,BA,O,O)", "triangle_property_angle_sum(1,OBA)"]} | |
6,928 | ChengQin_2023-06-04 | GeoQA-angle-10180 | 3 | 如图所示，∠CBE=56°，O是圆O的圆心。求∠COA的大小。 | As shown in the diagram, ∠CBE=56°, the center of circle O is O. Find the measure of ∠COA. | 6928.png | [
"Shape(DA,AO,OC,CD)",
"Shape(ODA,AD)",
"Shape(OCD,DC)",
"Shape(OA,AB,BC,CO)",
"Shape(OAB,BA)",
"Shape(OBC,CB)",
"Collinear(ABE)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(CBE),56)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(COA)) | 112 | [
"arc_property_circumference_angle_internal(1,OAC,B)",
"adjacent_complementary_angle(1,ABC,CBE)",
"arc_property_center_angle(1,OAC,O)"
] | {"START": ["arc_property_circumference_angle_internal(1,OAC,B)", "adjacent_complementary_angle(1,ABC,CBE)", "arc_property_center_angle(1,OAC,O)"]} | |
6,929 | ChengQin_2023-06-04 | GeoQA-angle-10181 | 4 | 如图所示，∠CAB=50°，△ABC的内切圆圆心为O。求∠COB的大小。 | As shown in the diagram, ∠CAB=50°, O is the incenter of triangle ABC. Find the measure of ∠COB. | 6929.png | [
"Shape(AB,BO,OA)",
"Shape(AO,OC,CA)",
"Shape(OB,BC,CO)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"IsIncenterOfTriangle(O,ABC)"
] | [] | Value(MeasureOfAngle(COB)) | 115 | [
"angle_addition(1,ABO,OBC)",
"angle_addition(1,BCO,OCA)",
"triangle_property_angle_sum(1,ABC)",
"triangle_property_angle_sum(1,OBC)"
] | {"START": ["angle_addition(1,ABO,OBC)", "angle_addition(1,BCO,OCA)", "triangle_property_angle_sum(1,ABC)", "triangle_property_angle_sum(1,OBC)"]} | |
6,930 | ChengQin_2023-06-04 | GeoQA-angle-10182 | 9 | 如图所示，∠DBA=63°，∠DCO=24°，⊙O的圆心为O，⊙O的直径为BA。求∠CDB的大小。 | As shown in the diagram, ∠DBA=63°, ∠DCO=24°, O is the center of ⊙O, BA is the diameter of ⊙O. Find the measure of ∠CDB. | 6930.png | [
"Shape(ODB,BD)",
"Shape(DB,BE,ED)",
"Shape(DE,EO,OA,OAD)",
"Shape(OBC,CE,EB)",
"Shape(EC,CO,OE)",
"Shape(OC,CA,AO)",
"Shape(OCA,AC)",
"Collinear(BEOA)",
"Collinear(DEC)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(DBA),63)",
"Equal(MeasureOfAngle(DCO),24)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BA,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 39 | [
"arc_property_circumference_angle_external(1,OAD,B)",
"arc_property_circumference_angle_external(1,OAD,C)",
"angle_addition(1,DCO,OCA)",
"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_triangl... | {"START": ["arc_property_circumference_angle_external(1,OAD,B)", "arc_property_circumference_angle_external(1,OAD,C)", "angle_addition(1,DCO,OCA)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circ... | |
6,931 | ChengQin_2023-06-04 | GeoQA-angle-10183 | 5 | 如图所示，∠BOC=56°，圆O的圆心为O，C是线段AB的中点。求∠ADB的大小。 | As shown in the diagram, ∠BOC=56°, O is the center of circle O, C is the midpoint of segment AB. Find the measure of ∠ADB. | 6931.png | [
"Shape(OPA,AP)",
"Shape(OBP,PB)",
"Shape(PA,AO,OB,BP)",
"Shape(OA,AC,CO)",
"Shape(OC,CB,BO)",
"Shape(CA,AD,DB,BC)",
"Shape(OAD,DA)",
"Shape(ODB,BD)",
"Collinear(ACB)",
"Cocircular(O,ADBP)"
] | [
"Equal(MeasureOfAngle(BOC),56)",
"IsCentreOfCircle(O,O)",
"IsMidpointOfLine(C,AB)"
] | [] | Value(MeasureOfAngle(ADB)) | 124 | [
"circle_property_chord_perpendicular_bisect_chord(2,O,OC,AB)",
"perpendicular_bisector_property_bisector(1,OC,AB)",
"angle_addition(1,BOC,COA)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_internal(1,OAB,D)"
] | {"START": ["circle_property_chord_perpendicular_bisect_chord(2,O,OC,AB)", "angle_addition(1,BOC,COA)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_internal(1,OAB,D)"], "circle_property_chord_perpendicular_bisect_chord(2,O,OC,AB)": ["perpendicular_bisector_property_bisector(1,OC,AB)"]} | |
6,932 | ChengQin_2023-06-04 | GeoQA-angle-10184 | 7 | 如图所示，∠CDA=120°，O是⊙O的圆心，AB是圆O的直径。求∠CEB的大小。 | As shown in the diagram, ∠CDA=120°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠CEB. | 6932.png | [
"Shape(OCD,DC)",
"Shape(OBC,CD)",
"Shape(ODE,EF,FD)",
"Shape(OEA,AG,GE)",
"Shape(EG,GF,FE)",
"Shape(FG,GH,HC,CF)",
"Shape(DF,FC,CD)",
"Shape(GA,AH,HG)",
"Shape(CH,HB,BC)",
"Shape(HA,AO,OB,BH)",
"Shape(OAB,BO,OA)",
"Collinear(AGFD)",
"Collinear(EGHB)",
"Collinear(EFC)",
"Collinear(AHC)",
... | [
"Equal(MeasureOfAngle(CDA),120)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CEB)) | 30 | [
"round_arc(1,OCA,OAC)",
"arc_property_circumference_angle_internal(1,OCA,D)",
"arc_property_circumference_angle_internal(1,OAC,B)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,BCA)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_... | {"START": ["round_arc(1,OCA,OAC)", "arc_property_circumference_angle_internal(1,OCA,D)", "arc_property_circumference_angle_internal(1,OAC,B)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,BCA)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle... | |
6,933 | ChengQin_2023-06-04 | GeoQA-angle-10185 | 2 | 如图所示，∠ABC=25°，圆O的圆心为O。求∠AOC的大小。 | As shown in the diagram, ∠ABC=25°, the center of circle O is O. Find the measure of ∠AOC. | 6933.png | [
"Shape(OAB,BA)",
"Shape(OCA,AC)",
"Shape(AB,BD,DA)",
"Shape(AD,DC,CA)",
"Shape(DO,OC,CD)",
"Shape(OD,DB,OBC,CO)",
"Collinear(ADO)",
"Collinear(BDC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(ABC),25)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOC)) | 50 | [
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_center_angle(1,OCA,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,B)", "arc_property_center_angle(1,OCA,O)"]} | |
6,934 | ChengQin_2023-06-04 | GeoQA-angle-10186 | 2 | 如图所示，∠BCA=43°，⊙O的圆心为O。求∠BOA的大小。 | As shown in the diagram, ∠BCA=43°, the center of circle O is O. Find the measure of ∠BOA. | 6934.png | [
"Shape(OCA,AC)",
"Shape(CA,AO,OB,BC)",
"Shape(OBC,CB)",
"Shape(OA,OAB,BO)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCA),43)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOA)) | 86 | [
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_center_angle(1,OAB,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OAB,C)", "arc_property_center_angle(1,OAB,O)"]} | |
6,935 | ChengQin_2023-06-04 | GeoQA-angle-10187 | 6 | 如图所示，∠AOC=60°，⊙O的圆心为O，圆O的直径为BC。求∠OAB的大小。 | As shown in the diagram, ∠AOC=60°, O is the center of circle O, BC is the diameter of circle O. Find the measure of ∠OAB. | 6935.png | [
"Shape(OAB,BA)",
"Shape(BO,OA,AB)",
"Shape(OBC,CO,OB)",
"Shape(OC,OCA,AO)",
"Collinear(BOC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(AOC),60)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BC,O)"
] | [] | Value(MeasureOfAngle(OAB)) | 30 | [
"arc_property_center_angle(1,OCA,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"isosceles_triangle_judgment_line_equal(1,OAB)",
"isosceles_triangle_property_angle_equal(1,OAB)"
] | {"START": ["arc_property_center_angle(1,OCA,O)", "arc_property_circumference_angle_external(1,OCA,B)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OA,O)"], "isosceles_triangle_judgment_line_equal(1,OAB)": ["isosceles_triangle_property_angle_equal(1,OAB)"], "radius_of_circ... | |
6,936 | ChengQin_2023-06-04 | GeoQA-angle-10188 | 6 | 如图所示，⊙O的圆心为O，圆O的直径为AB，D平分⌒OAB。求∠BCD的大小。 | As shown in the diagram, the center of ⊙O is O, the diameter of ⊙O is AB, D bisects arc OAB. Find the measure of ∠BCD. | 6936.png | [
"Shape(AO,OE,EC,OCA)",
"Shape(OBC,CB)",
"Shape(CE,EB,BC)",
"Shape(OA,OAD,DO)",
"Shape(OD,DE,EO)",
"Shape(ED,ODB,BE)",
"Collinear(AOEB)",
"Collinear(DEC)",
"Cocircular(O,ADBC)"
] | [
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsMidpointOfArc(D,OAB)"
] | [] | Value(MeasureOfAngle(BCD)) | 45 | [
"congruent_arc_judgment_length_equal(1,OAD,ODB)",
"congruent_arc_property_measure_equal(1,OAD,ODB)",
"arc_property_center_angle(1,OAD,O)",
"arc_property_center_angle(1,ODB,O)",
"adjacent_complementary_angle(1,BOD,DOA)",
"arc_property_circumference_angle_external(1,ODB,C)"
] | {"START": ["congruent_arc_judgment_length_equal(1,OAD,ODB)", "arc_property_center_angle(1,OAD,O)", "arc_property_center_angle(1,ODB,O)", "adjacent_complementary_angle(1,BOD,DOA)", "arc_property_circumference_angle_external(1,ODB,C)"], "congruent_arc_judgment_length_equal(1,OAD,ODB)": ["congruent_arc_property_measure_eq... | |
6,937 | ChengQin_2023-06-04 | GeoQA-angle-10189 | 4 | 如图所示，∠CDA=27°，O是⊙O的圆心，BA是⊙O的切线。求∠ABC的大小。 | As shown in the diagram, ∠CDA=27°, O is the center of circle O, BA is the tangent to circle O. Find the measure of ∠ABC. | 6937.png | [
"Shape(OC,OCD,DO)",
"Shape(OD,DE,EO)",
"Shape(OE,EC,CO)",
"Shape(DA,AE,ED)",
"Shape(EA,OAC,CE)",
"Shape(ODA,AD)",
"Shape(OAC,AB,BC)",
"Collinear(DEC)",
"Collinear(OEA)",
"Collinear(OCB)",
"Cocircular(O,ACD)"
] | [
"Equal(MeasureOfAngle(CDA),27)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(BA,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 36 | [
"arc_property_circumference_angle_external(1,OAC,D)",
"arc_property_center_angle(1,OAC,O)",
"tangent_of_circle_property_perpendicular(1,BA,O,O)",
"triangle_property_angle_sum(1,OAB)"
] | {"START": ["arc_property_circumference_angle_external(1,OAC,D)", "arc_property_center_angle(1,OAC,O)", "tangent_of_circle_property_perpendicular(1,BA,O,O)", "triangle_property_angle_sum(1,OAB)"]} | |
6,938 | ChengQin_2023-06-04 | GeoQA-angle-10190 | 9 | 如图所示，∠CAB=50°，O是圆O的圆心，BE⊥OE。求∠CBD的大小。 | As shown in the diagram, ∠CAB=50°, O is the center of ⊙O, BE⊥OE. Find the measure of ∠CBD. | 6938.png | [
"Shape(OAB,BA)",
"Shape(OCA,AC)",
"Shape(AB,BE,EO,OA)",
"Shape(AO,OE,EC,CA)",
"Shape(BD,DE,EB)",
"Shape(ED,DC,CE)",
"Shape(OBD,DB)",
"Shape(ODC,CD)",
"Collinear(BEC)",
"Collinear(OED)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(BE,OE)"
] | [] | Value(MeasureOfAngle(CBD)) | 25 | [
"arc_property_circumference_angle_internal(1,OBC,D)",
"circle_property_chord_perpendicular_bisect_chord(1,O,OE,BC)",
"vertical_angle(1,BEO,CED)",
"perpendicular_bisector_judgment_per_and_mid(1,DE,CB)",
"perpendicular_bisector_property_distance_equal(1,DE,CB)",
"arc_property_circumference_angle_external(1,... | {"START": ["arc_property_circumference_angle_internal(1,OBC,D)", "circle_property_chord_perpendicular_bisect_chord(1,O,OE,BC)", "vertical_angle(1,BEO,CED)", "arc_property_circumference_angle_external(1,OBC,A)", "triangle_property_angle_sum(1,DCB)"], "circle_property_chord_perpendicular_bisect_chord(1,O,OE,BC)": ["perpe... | |
6,939 | ChengQin_2023-06-04 | GeoQA-angle-10191 | 7 | 如图所示，∠ACB=70°，⊙O的圆心为O。求∠BAO的大小。 | As shown in the diagram, ∠ACB=70°, the center of ⊙O is O. Find the measure of ∠BAO. | 6939.png | [
"Shape(OBA,AB)",
"Shape(OCB,BC)",
"Shape(BA,AD,DB)",
"Shape(BD,DC,CB)",
"Shape(DA,AO,OD)",
"Shape(OA,OAC,CD,DO)",
"Collinear(ADC)",
"Collinear(BDO)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(ACB),70)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BAO)) | 20 | [
"arc_property_circumference_angle_external(1,OBA,C)",
"arc_property_center_angle(1,OBA,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OBA)",
"isosceles_triangle_property_angle_equal(1,OBA)",
"triangle_pro... | {"START": ["arc_property_circumference_angle_external(1,OBA,C)", "arc_property_center_angle(1,OBA,O)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,OBA)"], "isosceles_triangle_judgment_line_equal(1,OBA)": ["isosceles_triangle_property... | |
6,940 | ChengQin_2023-06-04 | GeoQA-angle-10192 | 4 | 如图所示，∠ACE=20°，⊙O的圆心为O，圆O的直径为AB。求∠BDE的大小。 | As shown in the diagram, ∠ACE=20°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BDE. | 6940.png | [
"Shape(OAC,CA)",
"Shape(OEA,AE)",
"Shape(ODE,ED)",
"Shape(OBD,DB)",
"Shape(AC,CF,FA)",
"Shape(AF,FG,GA)",
"Shape(AG,GE,EA)",
"Shape(FO,OB,BD,DG,GF)",
"Shape(GD,DE,EG)",
"Shape(OF,FC,OCB,BO)",
"Collinear(CFGE)",
"Collinear(AFOB)",
"Collinear(AGD)",
"Cocircular(O,ACBDE)"
] | [
"Equal(MeasureOfAngle(ACE),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BDE)) | 110 | [
"diameter_of_circle_property_right_angle(1,BDA,O)",
"arc_property_circumference_angle_external(1,OEA,C)",
"arc_property_circumference_angle_external(1,OEA,D)",
"angle_addition(1,BDA,ADE)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BDA,O)", "arc_property_circumference_angle_external(1,OEA,C)", "arc_property_circumference_angle_external(1,OEA,D)", "angle_addition(1,BDA,ADE)"]} | |
6,941 | ChengQin_2023-06-04 | GeoQA-angle-10195 | 2 | 如图所示，∠CAB=35°，⊙O的圆心为O，⊙O的直径为AB，CB是圆O的切线。求∠BCA的大小。 | As shown in the diagram, ∠CAB=35°, the center of ⊙O is O, AB is the diameter of ⊙O, the tangent to circle O is CB. Find the measure of ∠BCA. | 6941.png | [
"Shape(OAB,BO,OA)",
"Shape(ODA,AD)",
"Shape(AO,OB,OBD,DA)",
"Shape(OBD,BC,CD)",
"Collinear(ADC)",
"Collinear(AOB)",
"Cocircular(O,ABD)"
] | [
"Equal(MeasureOfAngle(CAB),35)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(CB,O)"
] | [] | Value(MeasureOfAngle(BCA)) | 55 | [
"tangent_of_circle_property_perpendicular(1,CB,O,O)",
"triangle_property_angle_sum(1,ABC)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,CB,O,O)", "triangle_property_angle_sum(1,ABC)"]} | |
6,942 | ChengQin_2023-06-04 | GeoQA-angle-10196 | 5 | 如图所示，∠DCB=Mul(2,MeasureOfAngle(BAD))°，O是⊙O的圆心，BE是圆O的直径。求∠DAE的大小。 | As shown in the diagram, ∠DCB=Mul(2,MeasureOfAngle(BAD))°, O is the center of ⊙O, the diameter of ⊙O is BE. Find the measure of ∠DAE. | 6942.png | [
"Shape(OBA,AB)",
"Shape(OCB,BC)",
"Shape(BA,AF,FO,OB)",
"Shape(BO,OF,FD,DC,CB)",
"Shape(ODC,CD)",
"Shape(FA,AE,EF)",
"Shape(FE,OED,DF)",
"Shape(OAE,EA)",
"Collinear(AFD)",
"Collinear(BOFE)",
"Cocircular(O,AEDCB)"
] | [
"Equal(MeasureOfAngle(DCB),Mul(2,MeasureOfAngle(BAD)))",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(BE,O)"
] | [] | Value(MeasureOfAngle(DAE)) | 30 | [
"round_arc(1,ODB,OBD)",
"arc_property_circumference_angle_internal(1,ODB,C)",
"arc_property_circumference_angle_internal(1,OBD,A)",
"diameter_of_circle_property_right_angle(1,BAE,O)",
"angle_addition(1,BAD,DAE)"
] | {"START": ["round_arc(1,ODB,OBD)", "arc_property_circumference_angle_internal(1,ODB,C)", "arc_property_circumference_angle_internal(1,OBD,A)", "diameter_of_circle_property_right_angle(1,BAE,O)", "angle_addition(1,BAD,DAE)"]} | |
6,943 | ChengQin_2023-06-04 | GeoQA-angle-10197 | 7 | 如图所示，∠COA=50°，⊙O的圆心为O，圆O的切线为AC。求∠ACD的大小。 | As shown in the diagram, ∠COA=50°, O is the center of ⊙O, AC is the tangent to circle O. Find the measure of ∠ACD. | 6943.png | [
"Shape(OC,OCD,DO)",
"Shape(OD,DC,CO)",
"Shape(ODC,CD)",
"Shape(DA,AC,ODC)",
"Collinear(ADO)",
"Cocircular(O,DC)"
] | [
"Equal(MeasureOfAngle(COA),50)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AC,O)"
] | [] | Value(MeasureOfAngle(ACD)) | 25 | [
"tangent_of_circle_property_perpendicular(2,AC,O,O)",
"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)",
"triangle_property_angle_sum(1,ODC)",
"angle_additi... | {"START": ["tangent_of_circle_property_perpendicular(2,AC,O,O)", "radius_of_circle_property_length_equal(1,OD,O)", "radius_of_circle_property_length_equal(1,OC,O)", "triangle_property_angle_sum(1,ODC)", "angle_addition(1,ACD,DCO)"], "isosceles_triangle_judgment_line_equal(1,ODC)": ["isosceles_triangle_property_angle_eq... | |
6,944 | ChengQin_2023-06-04 | GeoQA-angle-10198 | 8 | 如图所示，∠BPA=70°，O是⊙O的圆心，圆O的切线为PA，PB是⊙O的切线。求∠OBA的大小。 | As shown in the diagram, ∠BPA=70°, O is the center of circle O, PA is the tangent to circle O, PB is the tangent to ⊙O. Find the measure of ∠OBA. | 6944.png | [
"Shape(BO,OA,OAB)",
"Shape(OB,BA,AO)",
"Shape(OBA,AB)",
"Shape(BP,PA,OBA)",
"Cocircular(O,BA)"
] | [
"Equal(MeasureOfAngle(BPA),70)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PA,O)",
"IsTangentOfCircle(PB,O)"
] | [] | Value(MeasureOfAngle(OBA)) | 35 | [
"tangent_of_circle_property_perpendicular(1,PB,O,O)",
"tangent_of_circle_property_perpendicular(2,PA,O,O)",
"quadrilateral_property_angle_sum(1,OBPA)",
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OA,O)",
"isosceles_triangle_judgment_line_equal(1,OBA)",
"is... | {"START": ["tangent_of_circle_property_perpendicular(1,PB,O,O)", "tangent_of_circle_property_perpendicular(2,PA,O,O)", "quadrilateral_property_angle_sum(1,OBPA)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OA,O)", "triangle_property_angle_sum(1,OBA)"], "isosceles_triangl... | |
6,945 | ChengQin_2023-06-04 | GeoQA-angle-10199 | 6 | 如图所示，∠ABC=50°，O是⊙O的圆心，BA是圆O的切线，AD∥BO。求∠DCO的大小。 | As shown in the diagram, ∠ABC=50°, the center of circle O is O, the tangent to ⊙O is BA, AD∥BO. Find the measure of ∠DCO. | 6945.png | [
"Shape(OC,OCD,DE,EO)",
"Shape(OE,EC,CO)",
"Shape(DA,AE,ED)",
"Shape(EA,OAC,CE)",
"Shape(ODA,AD)",
"Shape(OAC,AB,BC)",
"Collinear(OCB)",
"Collinear(OEA)",
"Collinear(DEC)",
"Cocircular(O,ACD)"
] | [
"Equal(MeasureOfAngle(ABC),50)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(BA,O)",
"ParallelBetweenLine(AD,BO)"
] | [] | Value(MeasureOfAngle(DCO)) | 20 | [
"tangent_of_circle_property_perpendicular(1,BA,O,O)",
"triangle_property_angle_sum(1,OAB)",
"arc_property_center_angle(1,OAC,O)",
"arc_property_circumference_angle_external(1,OAC,D)",
"parallel_property_collinear_extend(3,OB,DA,C)",
"parallel_property_alternate_interior_angle(2,AD,CO)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,BA,O,O)", "triangle_property_angle_sum(1,OAB)", "arc_property_center_angle(1,OAC,O)", "arc_property_circumference_angle_external(1,OAC,D)", "parallel_property_collinear_extend(3,OB,DA,C)"], "parallel_property_collinear_extend(3,OB,DA,C)": ["parallel_property_altern... | |
6,946 | ChengQin_2023-06-04 | GeoQA-angle-10200 | 28 | 如图所示，AB=3，BC=4，⊙O的圆心为O，AG是圆O的切线，⊙O的切线为BH，FK垂直于OK，四边形ABCD是长方形。求直线EF的长度。 | As shown in the diagram, AB=3, BC=4, O is the center of ⊙O, AG is the tangent to ⊙O, the tangent to ⊙O is BH, FK is perpendicular to OK, ABCD is a rectangle. Find the length of line EF. | 6946.png | [
"Shape(AI,OEI,EA)",
"Shape(AG,OIG,IA)",
"Shape(GB,BH,OGH)",
"Shape(OG,OGH,HO)",
"Shape(OI,OIG,GO)",
"Shape(OEI,IO,OK,KE)",
"Shape(OFE,EK,KF)",
"Shape(FK,KO,OF)",
"Shape(OJF,FO,OJ)",
"Shape(OH,OHJ,JO)",
"Shape(OHJ,HC,CJ)",
"Shape(OJF,JC,CD,DF)",
"Collinear(AEKFD)",
"Collinear(AIOJC)",
"Co... | [
"Equal(LengthOfLine(AB),3)",
"Equal(LengthOfLine(BC),4)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AG,O)",
"IsTangentOfCircle(BH,O)",
"PerpendicularBetweenLine(FK,OK)",
"Rectangle(ABCD)"
] | [] | Value(LengthOfLine(EF)) | 6*sqrt(7)/7 | [
"radius_of_circle_property_length_equal(1,OG,O)",
"radius_of_circle_property_length_equal(1,OH,O)",
"tangent_of_circle_property_perpendicular(2,AG,O,O)",
"tangent_of_circle_property_perpendicular(2,BH,O,O)",
"parallel_judgment_ipsilateral_internal_angle(1,BG,HO)",
"flat_angle(1,AGB)",
"angle_addition(1,... | {"START": ["radius_of_circle_property_length_equal(1,OG,O)", "radius_of_circle_property_length_equal(1,OH,O)", "tangent_of_circle_property_perpendicular(2,AG,O,O)", "tangent_of_circle_property_perpendicular(2,BH,O,O)", "flat_angle(1,AGB)", "angle_addition(1,AGO,OGB)", "line_addition(1,AG,GB)", "radius_of_circle_propert... | |
6,947 | ChengQin_2023-06-04 | GeoQA-angle-10201 | 5 | 如图所示，∠ABC=25°，O是圆O的圆心，圆O的切线为AB。求∠CDB的大小。 | As shown in the diagram, ∠ABC=25°, the center of circle O is O, AB is the tangent to circle O. Find the measure of ∠CDB. | 6947.png | [
"Shape(OBC,CB)",
"Shape(OCA,AC)",
"Shape(CA,AE,EC)",
"Shape(CE,EO,OC)",
"Shape(CO,OB,BC)",
"Shape(OAD,DE,EA)",
"Shape(OE,ED,DB,BO)",
"Shape(ODB,BD)",
"Collinear(CED)",
"Collinear(AEOB)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(ABC),25)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 65 | [
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_center_angle(1,OCA,O)",
"adjacent_complementary_angle(1,AOC,COB)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,D)"
] | {"START": ["arc_property_circumference_angle_external(1,OCA,B)", "arc_property_center_angle(1,OCA,O)", "adjacent_complementary_angle(1,AOC,COB)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,D)"]} | |
6,948 | ChengQin_2023-06-04 | GeoQA-angle-10202 | 6 | 如图所示，AB=sqrt(2)，∠CAB=30°，⊙O的半径为1，圆O的圆心为O。求∠ABC的大小。 | As shown in the diagram, AB=sqrt(2), ∠CAB=30°, the radius of ⊙O is 1, O is the center of circle O. Find the measure of ∠ABC. | 6948.png | [
"Shape(OA,AD,DO)",
"Shape(OD,DC,OCA,AO)",
"Shape(AB,BD,DA)",
"Shape(DB,BC,CD)",
"Shape(OAB,BAD)",
"Shape(OBC,CB)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(LengthOfLine(AB),sqrt(2))",
"Equal(MeasureOfAngle(CAB),30)",
"Equal(RadiusOfCircle(O),1)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 105 | [
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"right_triangle_judgment_pythagorean_inverse(1,BOA)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"triangle_property_angle_sum(1,CAB)"
] | {"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,CAB)"], "radius_of_circle_property_length_equal(1,OA,O)": ["right_triangle_judgment_py... | |
6,949 | ChengQin_2023-06-04 | GeoQA-angle-10203 | 4 | 如图所示，∠CBA=70°，O是⊙O的圆心，AP⊥DP。求∠DEB的大小。 | As shown in the diagram, ∠CBA=70°, the center of circle O is O, AP⊥DP. Find the measure of ∠DEB. | 6949.png | [
"Shape(ODE,ED)",
"Shape(OBD,DF,FB)",
"Shape(DE,EF,FD)",
"Shape(FP,PB,BF)",
"Shape(OEA,AP,PF,FE)",
"Shape(OAC,CP,PA)",
"Shape(PC,CB,BP)",
"Shape(OCB,BC)",
"Collinear(DFPC)",
"Collinear(APB)",
"Collinear(EFB)",
"Cocircular(O,ACBDE)"
] | [
"Equal(MeasureOfAngle(CBA),70)",
"IsCentreOfCircle(O,O)",
"PerpendicularBetweenLine(AP,DP)"
] | [] | Value(MeasureOfAngle(DEB)) | 20 | [
"vertical_angle(1,APD,BPC)",
"triangle_property_angle_sum(1,BPC)",
"arc_property_circumference_angle_external(1,OBD,C)",
"arc_property_circumference_angle_external(1,OBD,E)"
] | {"START": ["vertical_angle(1,APD,BPC)", "triangle_property_angle_sum(1,BPC)", "arc_property_circumference_angle_external(1,OBD,C)", "arc_property_circumference_angle_external(1,OBD,E)"]} | |
6,950 | ChengQin_2023-06-04 | GeoQA-angle-10204 | 6 | 如图所示，∠OCA=26°，⊙O的圆心为O，AB是圆O的直径。求∠COB的大小。 | As shown in the diagram, ∠OCA=26°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠COB. | 6950.png | [
"Shape(OAB,BO,OA)",
"Shape(OC,CA,AO)",
"Shape(OB,OBC,CO)",
"Shape(OCA,AC)",
"Collinear(BOA)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(OCA),26)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(COB)) | 52 | [
"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)",
"adjacent_complementary_angle(1,AOC,COB)"
] | {"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)", "adjacent_complementary_angle(1,AOC,COB)"], "isosceles_triangle_judgment_line_equal(1,OCA)": ["isosceles_triangle_property_angle_equal(1,OCA)"], "radius_of_circle_property... | |
6,951 | ChengQin_2023-06-04 | GeoQA-angle-10205 | 6 | 如图所示，∠COB=60°，O是圆O的圆心，AB是⊙O的直径。求∠OCA的大小。 | As shown in the diagram, ∠COB=60°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠OCA. | 6951.png | [
"Shape(OCA,AC)",
"Shape(AO,OC,CA)",
"Shape(OAB,BO,OA)",
"Shape(OBC,CO,OB)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(COB),60)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(OCA)) | 30 | [
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,A)",
"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)"
] | {"START": ["arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,A)", "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)"], "radius_of_circ... | |
6,952 | ChengQin_2023-06-04 | GeoQA-angle-10206 | 2 | 如图所示，∠CAB=54°，圆O的圆心为O。求∠COB的大小。 | As shown in the diagram, ∠CAB=54°, the center of ⊙O is O. Find the measure of ∠COB. | 6952.png | [
"Shape(OCA,AC)",
"Shape(OAB,BC)",
"Shape(AB,BO,OC,CA)",
"Shape(OBC,CO,OB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),54)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(COB)) | 108 | [
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_center_angle(1,OBC,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OBC,A)", "arc_property_center_angle(1,OBC,O)"]} | |
6,953 | ChengQin_2023-06-04 | GeoQA-angle-10207 | 4 | 如图所示，∠DCE=64°，⊙O的圆心为O。求∠DOB的大小。 | As shown in the diagram, ∠DCE=64°, the center of ⊙O is O. Find the measure of ∠DOB. | 6953.png | [
"Shape(ODA,AD)",
"Shape(OAB,BA)",
"Shape(AB,BO,OD,DA)",
"Shape(OBC,CB)",
"Shape(OCD,DC)",
"Shape(OB,BC,CD,DO)",
"Collinear(BCE)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(DCE),64)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(DOB)) | 128 | [
"flat_angle(1,BCE)",
"angle_addition(1,BCD,DCE)",
"arc_property_circumference_angle_internal(1,OBD,C)",
"arc_property_center_angle(1,OBD,O)"
] | {"START": ["flat_angle(1,BCE)", "angle_addition(1,BCD,DCE)", "arc_property_circumference_angle_internal(1,OBD,C)", "arc_property_center_angle(1,OBD,O)"]} | |
6,954 | ChengQin_2023-06-04 | GeoQA-angle-10208 | 3 | 如图所示，∠AOC=90°，∠BAC=30°，O是⊙O的圆心。求∠AOB的大小。 | As shown in the diagram, ∠AOC=90°, ∠BAC=30°, the center of ⊙O is O. Find the measure of ∠AOB. | 6954.png | [
"Shape(OBA,AB)",
"Shape(OCB,BC)",
"Shape(AD,DB,BA)",
"Shape(BD,DC,CB)",
"Shape(AO,OD,DA)",
"Shape(DO,OC,CD)",
"Shape(OAC,CO,OA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ACB)"
] | [
"Equal(MeasureOfAngle(AOC),90)",
"Equal(MeasureOfAngle(BAC),30)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(AOB)) | 30 | [
"arc_property_circumference_angle_external(1,OCB,A)",
"arc_property_center_angle(1,OCB,O)",
"angle_addition(1,AOB,BOC)"
] | {"START": ["arc_property_circumference_angle_external(1,OCB,A)", "arc_property_center_angle(1,OCB,O)", "angle_addition(1,AOB,BOC)"]} | |
6,955 | ChengQin_2023-06-04 | GeoQA-angle-10209 | 7 | 如图所示，CD=Mul(2,LengthOfLine(OE))，O是圆O的圆心，AB是⊙O的直径，CE垂直于OE。求∠DCB的大小。 | As shown in the diagram, CD=Mul(2,LengthOfLine(OE)), the center of circle O is O, the diameter of ⊙O is AB, CE⊥OE. Find the measure of ∠DCB. | 6955.png | [
"Shape(AO,OD,ODA)",
"Shape(OAC,CE,EO,OA)",
"Shape(OE,ED,DO)",
"Shape(EC,CB,BE)",
"Shape(OCB,BC)",
"Shape(EB,OBD,DE)",
"Collinear(CED)",
"Collinear(AOEB)",
"Cocircular(O,ACBD)"
] | [
"Equal(LengthOfLine(CD),Mul(2,LengthOfLine(OE)))",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"PerpendicularBetweenLine(CE,OE)"
] | [] | Value(MeasureOfAngle(DCB)) | 45/2 | [
"circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)",
"line_addition(1,CE,ED)",
"isosceles_triangle_judgment_line_equal(1,EDO)",
"isosceles_triangle_property_angle_equal(1,EDO)",
"triangle_property_angle_sum(1,EDO)",
"arc_property_center_angle(1,OBD,O)",
"arc_property_circumference_angle_extern... | {"START": ["circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)", "line_addition(1,CE,ED)", "triangle_property_angle_sum(1,EDO)", "arc_property_center_angle(1,OBD,O)", "arc_property_circumference_angle_external(1,OBD,C)"], "circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)": ["isosceles_triangle_judgm... | |
6,956 | ChengQin_2023-06-04 | GeoQA-angle-10210 | 5 | 如图所示，∠APB=70°，⊙O的圆心为O，⊙O的切线为PA，⊙O的切线为PB。求∠ACB的大小。 | As shown in the diagram, ∠APB=70°, the center of circle O is O, the tangent to ⊙O is PA, PB is the tangent to circle O. Find the measure of ∠ACB. | 6956.png | [
"Shape(AP,PB,OCB,OAC)",
"Shape(OAC,CA)",
"Shape(OCB,BC)",
"Shape(AC,CB,BO,OA)",
"Shape(AO,OD,DA)",
"Shape(ODA,AD)",
"Shape(OB,BD,DO)",
"Shape(OBD,DB)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(APB),70)",
"IsCentreOfCircle(O,O)",
"IsTangentOfCircle(PA,O)",
"IsTangentOfCircle(PB,O)"
] | [] | Value(MeasureOfAngle(ACB)) | 125 | [
"tangent_of_circle_property_perpendicular(1,PA,O,O)",
"tangent_of_circle_property_perpendicular(2,PB,O,O)",
"quadrilateral_property_angle_sum(1,APBO)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_internal(1,OAB,C)"
] | {"START": ["tangent_of_circle_property_perpendicular(1,PA,O,O)", "tangent_of_circle_property_perpendicular(2,PB,O,O)", "quadrilateral_property_angle_sum(1,APBO)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_internal(1,OAB,C)"]} | |
6,957 | ChengQin_2023-06-04 | GeoQA-angle-10211 | 4 | 如图所示，AB=CD，∠BOA=42°，圆O的圆心为O。求∠DEC的大小。 | As shown in the diagram, AB=CD, ∠BOA=42°, the center of circle O is O. Find the measure of ∠DEC. | 6957.png | [
"Shape(OF,FE,OEA,AO)",
"Shape(OAB,BA)",
"Shape(OA,AB,BO)",
"Shape(OB,OBC,CO)",
"Shape(OC,CF,FO)",
"Shape(FC,CD,DF)",
"Shape(EF,FD,DE)",
"Shape(ODE,ED)",
"Shape(OCD,DC)",
"Collinear(OFD)",
"Collinear(CFE)",
"Cocircular(O,ABCDE)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(CD))",
"Equal(MeasureOfAngle(BOA),42)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(DEC)) | 21 | [
"congruent_arc_judgment_chord_equal(1,OAB,OCD)",
"congruent_arc_property_measure_equal(1,OAB,OCD)",
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OCD,E)"
] | {"START": ["congruent_arc_judgment_chord_equal(1,OAB,OCD)", "arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OCD,E)"], "congruent_arc_judgment_chord_equal(1,OAB,OCD)": ["congruent_arc_property_measure_equal(1,OAB,OCD)"]} | |
6,958 | ChengQin_2023-06-04 | GeoQA-angle-10212 | 11 | 如图所示，OA=3，三角形ABC为等边三角形，⊙O的圆心为O，圆O的直径为AD。求弧OBD的长度。 | As shown in the diagram, OA=3, △ABC is an equilateral △, O is the center of circle O, the diameter of ⊙O is AD. Find the length of ⌒OBD. | 6958.png | [
"Shape(OAB,BA)",
"Shape(OA,AB,BO)",
"Shape(AO,OE,EC,CA)",
"Shape(OCA,AC)",
"Shape(OB,BE,EO)",
"Shape(OBD,DB)",
"Shape(BD,DE,EB)",
"Shape(ODC,CE,ED)",
"Collinear(AOED)",
"Collinear(BEC)",
"Cocircular(O,ABDC)"
] | [
"Equal(LengthOfLine(OA),3)",
"EquilateralTriangle(ABC)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AD,O)"
] | [] | Value(LengthOfArc(OBD)) | pi | [
"equilateral_triangle_property_angle(1,CAB)",
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_circumference_angle_external(1,OAB,D)",
"radius_of_circle_property_length_equal(1,OB,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"isosceles_triangle_judgment_line_equal(1,OBD)",
"... | {"START": ["equilateral_triangle_property_angle(1,CAB)", "arc_property_circumference_angle_external(1,OAB,C)", "arc_property_circumference_angle_external(1,OAB,D)", "radius_of_circle_property_length_equal(1,OB,O)", "radius_of_circle_property_length_equal(1,OD,O)", "triangle_property_angle_sum(1,OBD)", "radius_of_circle... | |
6,959 | ChengQin_2023-06-04 | GeoQA-angle-10213 | 3 | 如图所示，O是圆O的圆心，四边形ABCD是正方形。求∠CPB的大小。 | As shown in the diagram, the center of circle O is O, ABCD is a square. Find the measure of ∠CPB. | 6959.png | [
"Shape(ODA,AD)",
"Shape(DA,AO,OD)",
"Shape(OCD,DC)",
"Shape(DO,OC,CD)",
"Shape(OAP,PE,EA)",
"Shape(AE,EF,FO,OA)",
"Shape(OPB,BP)",
"Shape(PB,BE,EP)",
"Shape(OF,FC,CO)",
"Shape(EB,BF,FE)",
"Shape(FB,BC,CF)",
"Shape(OBC,CB)",
"Collinear(AOC)",
"Collinear(AEB)",
"Collinear(BFOD)",
"Collin... | [
"IsCentreOfCircle(O,O)",
"Square(ABCD)"
] | [] | Value(MeasureOfAngle(CPB)) | 45 | [
"kite_property_diagonal_perpendicular_bisection(1,BCDA,O)",
"arc_property_center_angle(1,OBC,O)",
"arc_property_circumference_angle_external(1,OBC,P)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,BCDA,O)", "arc_property_center_angle(1,OBC,O)", "arc_property_circumference_angle_external(1,OBC,P)"]} | |
6,960 | ChengQin_2023-06-04 | GeoQA-angle-10214 | 10 | 如图所示，∠CAE=14°，⊙O的圆心为O，AB是⊙O的直径，D是⌒OCA的中点。求∠BEC的大小。 | As shown in the diagram, ∠CAE=14°, O is the center of ⊙O, AB is the diameter of circle O, D is the midpoint of arc OCA. Find the measure of ∠BEC. | 6960.png | [
"Shape(ODA,AD)",
"Shape(OCD,DC)",
"Shape(DA,AC,CD)",
"Shape(CA,AO,OB,BC)",
"Shape(OBC,CB)",
"Shape(OBC,BE,EC)",
"Shape(OA,OAB,BO)",
"Collinear(AOBE)",
"Collinear(DCE)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(CAE),14)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsMidpointOfArc(D,OCA)"
] | [] | Value(MeasureOfAngle(BEC)) | 24 | [
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,BCA)",
"arc_property_circumference_angle_external(1,OCA,B)",
"congruent_arc_judgment_length_equal(1,ODA,OCD)",
"congruent_arc_property_measure_equal(1,ODA,OCD)",
"arc_addition_measure(1,OCD,ODA)",
"arc_property_circumfere... | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,BCA)", "arc_property_circumference_angle_external(1,OCA,B)", "congruent_arc_judgment_length_equal(1,ODA,OCD)", "arc_addition_measure(1,OCD,ODA)", "arc_property_circumference_angle_external(1,ODA,C)", "triangle_property_angle_s... | |
6,961 | ChengQin_2023-06-04 | GeoQA-angle-10215 | 4 | 如图所示，∠ACD=40°，⊙O的圆心为O，⊙O的直径为AB。求∠DAB的大小。 | As shown in the diagram, ∠ACD=40°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠DAB. | 6961.png | [
"Shape(ODA,AD)",
"Shape(OCD,DC)",
"Shape(CD,DE,EC)",
"Shape(DA,AE,ED)",
"Shape(OBC,CE,EB)",
"Shape(EA,AO,OB,BE)",
"Shape(OA,OAB,BO)",
"Collinear(DEB)",
"Collinear(AEC)",
"Collinear(AOB)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(ACD),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DAB)) | 50 | [
"diameter_of_circle_property_right_angle(1,BDA,O)",
"arc_property_circumference_angle_external(1,ODA,C)",
"arc_property_circumference_angle_external(1,ODA,B)",
"triangle_property_angle_sum(1,DAB)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BDA,O)", "arc_property_circumference_angle_external(1,ODA,C)", "arc_property_circumference_angle_external(1,ODA,B)", "triangle_property_angle_sum(1,DAB)"]} | |
6,962 | ChengQin_2023-06-04 | GeoQA-angle-10216 | 5 | 如图所示，∠BOA=60°，∠OBC=55°，圆O的圆心为O。求∠OAC的大小。 | As shown in the diagram, ∠BOA=60°, ∠OBC=55°, O is the center of circle O. Find the measure of ∠OAC. | 6962.png | [
"Shape(OCA,AO,OC)",
"Shape(OA,AD,DO)",
"Shape(OD,DC,CO)",
"Shape(DB,BC,CD)",
"Shape(OBC,CB)",
"Shape(OAB,BD,DA)",
"Collinear(ADC)",
"Collinear(ODB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BOA),60)",
"Equal(MeasureOfAngle(OBC),55)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(OAC)) | 25 | [
"arc_property_center_angle(1,OAB,O)",
"arc_property_circumference_angle_external(1,OAB,C)",
"triangle_property_angle_sum(1,DBC)",
"vertical_angle(1,ADO,CDB)",
"triangle_property_angle_sum(1,OAD)"
] | {"START": ["arc_property_center_angle(1,OAB,O)", "arc_property_circumference_angle_external(1,OAB,C)", "triangle_property_angle_sum(1,DBC)", "vertical_angle(1,ADO,CDB)", "triangle_property_angle_sum(1,OAD)"]} | |
6,963 | ChengQin_2023-06-04 | GeoQA-angle-10217 | 4 | 如图所示，∠BOD=140°，⊙O的圆心为O，AB是圆O的直径。求∠DCA的大小。 | As shown in the diagram, ∠BOD=140°, the center of circle O is O, AB is the diameter of circle O. Find the measure of ∠DCA. | 6963.png | [
"Shape(OCA,AC)",
"Shape(CA,AE,EC)",
"Shape(CE,EO,OB,BC)",
"Shape(OBC,CB)",
"Shape(OAD,DE,EA)",
"Shape(OE,ED,DO)",
"Shape(OD,ODB,BO)",
"Collinear(AEOB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(BOD),140)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DCA)) | 20 | [
"diameter_of_circle_property_right_angle(1,BCA,O)",
"arc_property_center_angle(1,ODB,O)",
"arc_property_circumference_angle_external(1,ODB,C)",
"angle_addition(1,BCD,DCA)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)", "arc_property_center_angle(1,ODB,O)", "arc_property_circumference_angle_external(1,ODB,C)", "angle_addition(1,BCD,DCA)"]} | |
6,964 | ChengQin_2023-06-04 | GeoQA-angle-10218 | 4 | 如图所示，∠BCD=30°，O是圆O的圆心，⊙O的直径为AB。求∠DOA的大小。 | As shown in the diagram, ∠BCD=30°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DOA. | 6964.png | [
"Shape(OCA,AO,OE,EC)",
"Shape(OBC,CB)",
"Shape(BC,CE,EB)",
"Shape(OA,OAD,DO)",
"Shape(OD,DE,EO)",
"Shape(ED,ODB,BE)",
"Collinear(AOEB)",
"Collinear(CED)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(BCD),30)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(DOA)) | 120 | [
"arc_property_circumference_angle_external(1,ODB,C)",
"arc_property_center_angle(1,ODB,O)",
"flat_angle(1,BOA)",
"angle_addition(1,BOD,DOA)"
] | {"START": ["arc_property_circumference_angle_external(1,ODB,C)", "arc_property_center_angle(1,ODB,O)", "flat_angle(1,BOA)", "angle_addition(1,BOD,DOA)"]} | |
6,965 | ChengQin_2023-06-04 | GeoQA-angle-10219 | 2 | 如图所示，∠CAB=20°，O是⊙O的圆心，AB是⊙O的直径。求∠ABC的大小。 | As shown in the diagram, ∠CAB=20°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠ABC. | 6965.png | [
"Shape(OCA,AC)",
"Shape(OBC,CB)",
"Shape(CA,AO,OB,BC)",
"Shape(OAB,BO,OA)",
"Collinear(AOB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ABC)) | 70 | [
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,BCA)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,BCA)"]} | |
6,966 | ChengQin_2023-06-04 | GeoQA-angle-10220 | 2 | 如图所示，∠BCA=38°，⊙O的圆心为O。求∠BOA的大小。 | As shown in the diagram, ∠BCA=38°, the center of circle O is O. Find the measure of ∠BOA. | 6966.png | [
"Shape(OCA,AO,OD,DC)",
"Shape(OA,AD,DO)",
"Shape(DA,OAB,BD)",
"Shape(DB,BC,CD)",
"Shape(OBC,CB)",
"Collinear(ODB)",
"Collinear(ADC)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(BCA),38)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BOA)) | 76 | [
"arc_property_circumference_angle_external(1,OAB,C)",
"arc_property_center_angle(1,OAB,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OAB,C)", "arc_property_center_angle(1,OAB,O)"]} | |
6,967 | ChengQin_2023-06-04 | GeoQA-angle-10221 | 2 | 如图所示，∠CAB=40°，⊙O的圆心为O，△ABC的外心为O。求∠COB的大小。 | As shown in the diagram, ∠CAB=40°, the center of ⊙O is O, the center of the circumscribed circle of triangle ABC is O. Find the measure of ∠COB. | 6967.png | [
"Shape(OAB,BA)",
"Shape(AB,BO,OC,CA)",
"Shape(OCA,AC)",
"Shape(OB,BC,CO)",
"Shape(OBC,CB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),40)",
"IsCentreOfCircle(O,O)",
"IsCircumcenterOfTriangle(O,ABC)"
] | [] | Value(MeasureOfAngle(COB)) | 80 | [
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_center_angle(1,OBC,O)"
] | {"START": ["arc_property_circumference_angle_external(1,OBC,A)", "arc_property_center_angle(1,OBC,O)"]} | |
6,968 | ChengQin_2023-06-04 | GeoQA-angle-10222 | 2 | 如图所示，∠DAB=80°。求∠BCD的大小。 | As shown in the diagram, ∠DAB=80°. Find the measure of ∠BCD. | 6968.png | [
"Shape(ODA,AD)",
"Shape(OCD,DC)",
"Shape(OAB,BA)",
"Shape(OBC,CB)",
"Shape(DA,AB,BC,CD)",
"Cocircular(O,ABCD)"
] | [
"Equal(MeasureOfAngle(DAB),80)"
] | [] | Value(MeasureOfAngle(BCD)) | 100 | [
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_internal(1,OBD,C)"
] | {"START": ["arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_internal(1,OBD,C)"]} | |
6,969 | ChengQin_2023-06-04 | GeoQA-angle-10223 | 4 | 如图所示，∠ABC=20°，圆O的圆心为O，圆O的直径为CD。求∠DCA的大小。 | As shown in the diagram, ∠ABC=20°, O is the center of circle O, CD is the diameter of circle O. Find the measure of ∠DCA. | 6969.png | [
"Shape(OAD,DA)",
"Shape(ODB,BE,EO,OD)",
"Shape(AD,DO,OE,EA)",
"Shape(AE,EC,CA)",
"Shape(OCA,AC)",
"Shape(EB,BC,CE)",
"Shape(OBC,CB)",
"Collinear(DOEC)",
"Collinear(BEA)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(ABC),20)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(CD,O)"
] | [] | Value(MeasureOfAngle(DCA)) | 70 | [
"diameter_of_circle_property_right_angle(1,CAD,O)",
"arc_property_circumference_angle_external(1,OCA,B)",
"arc_property_circumference_angle_external(1,OCA,D)",
"triangle_property_angle_sum(1,CAD)"
] | {"START": ["diameter_of_circle_property_right_angle(1,CAD,O)", "arc_property_circumference_angle_external(1,OCA,B)", "arc_property_circumference_angle_external(1,OCA,D)", "triangle_property_angle_sum(1,CAD)"]} | |
6,970 | ChengQin_2023-06-04 | GeoQA-angle-10224 | 6 | 如图所示，∠CAD=20°，△ABC为等边△。求∠DCA的大小。 | As shown in the diagram, ∠CAD=20°, △ABC is an equilateral △. Find the measure of ∠DCA. | 6970.png | [
"Shape(OAB,BA)",
"Shape(AB,BE,EA)",
"Shape(AE,EC,CA)",
"Shape(OCA,AC)",
"Shape(EB,BD,DE)",
"Shape(CE,ED,DC)",
"Shape(OBD,DB)",
"Shape(ODC,CD)",
"Collinear(BEC)",
"Collinear(AED)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(CAD),20)",
"EquilateralTriangle(ABC)"
] | [] | Value(MeasureOfAngle(DCA)) | 100 | [
"equilateral_triangle_property_angle(1,ABC)",
"angle_addition(1,CAD,DAB)",
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,OBD,C)",
"equilateral_triangle_property_angle(1,CAB)",
"angle_addition(1,DCB,BCA)"
] | {"START": ["equilateral_triangle_property_angle(1,ABC)", "angle_addition(1,CAD,DAB)", "arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,OBD,C)", "equilateral_triangle_property_angle(1,CAB)", "angle_addition(1,DCB,BCA)"]} | |
6,971 | ChengQin_2023-06-04 | GeoQA-angle-10225 | 10 | 如图所示，∠CAB=50°，O是圆O的圆心，E平分线段BC。求∠CBD的大小。 | As shown in the diagram, ∠CAB=50°, O is the center of circle O, E bisects segment BC. Find the measure of ∠CBD. | 6971.png | [
"Shape(OAB,BF,FA)",
"Shape(AF,FO,OE,EC,CA)",
"Shape(FB,BE,EO,OF)",
"Shape(OCA,AC)",
"Shape(OBD,DB)",
"Shape(EB,BD,DE)",
"Shape(ED,DC,CE)",
"Shape(ODC,CD)",
"Collinear(AFB)",
"Collinear(FOED)",
"Collinear(BEC)",
"Cocircular(O,ABDC)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"IsCentreOfCircle(O,O)",
"IsMidpointOfLine(E,BC)"
] | [] | Value(MeasureOfAngle(CBD)) | 25 | [
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_internal(1,OBC,D)",
"circle_property_chord_perpendicular_bisect_chord(2,O,OE,BC)",
"flat_angle(1,OED)",
"angle_addition(1,OEC,CED)",
"perpendicular_bisector_judgment_per_and_mid(1,DE,CB)",
"perpendicular_bisector_pro... | {"START": ["arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_internal(1,OBC,D)", "circle_property_chord_perpendicular_bisect_chord(2,O,OE,BC)", "flat_angle(1,OED)", "angle_addition(1,OEC,CED)", "triangle_property_angle_sum(1,DCB)"], "angle_addition(1,OEC,CED)": ["perpendicular_bise... | |
6,972 | ChengQin_2023-06-04 | GeoQA-angle-10226 | 4 | 如图所示，∠DCB=44°，O是圆O的圆心，⊙O的直径为AB。求∠ABD的大小。 | As shown in the diagram, ∠DCB=44°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠ABD. | 6972.png | [
"Shape(ODA,AD)",
"Shape(DA,AO,OE,ED)",
"Shape(OBD,DB)",
"Shape(DE,EB,BD)",
"Shape(OA,OAC,CE,EO)",
"Shape(EC,CB,BE)",
"Shape(OCB,BC)",
"Collinear(AOEB)",
"Collinear(DEC)",
"Cocircular(O,ACBD)"
] | [
"Equal(MeasureOfAngle(DCB),44)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(ABD)) | 46 | [
"diameter_of_circle_property_right_angle(1,BDA,O)",
"arc_property_circumference_angle_external(1,OBD,A)",
"arc_property_circumference_angle_external(1,OBD,C)",
"triangle_property_angle_sum(1,ABD)"
] | {"START": ["diameter_of_circle_property_right_angle(1,BDA,O)", "arc_property_circumference_angle_external(1,OBD,A)", "arc_property_circumference_angle_external(1,OBD,C)", "triangle_property_angle_sum(1,ABD)"]} | |
6,973 | ChengQin_2023-06-04 | GeoQA-angle-10227 | 10 | 如图所示，BC=CD，∠CBA=52°，圆O的圆心为O。求∠BAC的大小。 | As shown in the diagram, BC=CD, ∠CBA=52°, O is the center of ⊙O. Find the measure of ∠BAC. | 6973.png | [
"Shape(OBA,AD,DO,OB)",
"Shape(OD,DB,BO)",
"Shape(BD,DC,CB)",
"Shape(OCB,BC)",
"Shape(DA,AC,CD)",
"Shape(OAC,CA)",
"Collinear(ADB)",
"Collinear(ODC)",
"Cocircular(O,ACB)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(CD))",
"Equal(MeasureOfAngle(CBA),52)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BAC)) | 14 | [
"isosceles_triangle_judgment_line_equal(1,CBD)",
"isosceles_triangle_property_angle_equal(1,CBD)",
"triangle_property_angle_sum(1,CBD)",
"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_triangl... | {"START": ["isosceles_triangle_judgment_line_equal(1,CBD)", "triangle_property_angle_sum(1,CBD)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,OCB)", "arc_property_center_angle(1,OCB,O)", "arc_property_circumference_angle_external(1,O... | |
6,974 | ChengQin_2023-06-04 | GeoQA-angle-10228 | 12 | 如图所示，∠FOA=40°，圆O的圆心为O，圆O的直径为AB，E是弧OBF的中点。求∠CFO的大小。 | As shown in the diagram, ∠FOA=40°, O is the center of ⊙O, the diameter of ⊙O is AB, E is the midpoint of ⌒OBF. Find the measure of ∠CFO. | 6974.png | [
"Shape(OEA,AC,CE)",
"Shape(EC,CO,OE)",
"Shape(EO,OB,BE)",
"Shape(OBE,EB)",
"Shape(OAF,FC,CA)",
"Shape(CF,FO,OC)",
"Shape(OF,OFB,BO)",
"Collinear(ACOB)",
"Collinear(FCE)",
"Cocircular(O,AFBE)"
] | [
"Equal(MeasureOfAngle(FOA),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsMidpointOfArc(E,OBF)"
] | [] | Value(MeasureOfAngle(CFO)) | 35 | [
"congruent_arc_judgment_length_equal(1,OEF,OBE)",
"congruent_arc_property_measure_equal(1,OEF,OBE)",
"arc_property_center_angle(1,OEF,O)",
"arc_property_center_angle(1,OBE,O)",
"flat_angle(1,AOB)",
"angle_addition(1,AOE,EOB)",
"angle_addition(1,FOA,AOE)",
"radius_of_circle_property_length_equal(1,OE,O... | {"START": ["congruent_arc_judgment_length_equal(1,OEF,OBE)", "arc_property_center_angle(1,OEF,O)", "arc_property_center_angle(1,OBE,O)", "flat_angle(1,AOB)", "angle_addition(1,AOE,EOB)", "angle_addition(1,FOA,AOE)", "radius_of_circle_property_length_equal(1,OE,O)", "radius_of_circle_property_length_equal(1,OF,O)", "tri... | |
6,975 | ChengQin_2023-06-04 | GeoQA-angle-10229 | 7 | 如图所示，∠CAB=64°，⊙O的圆心为O。求∠BCO的大小。 | As shown in the diagram, ∠CAB=64°, the center of ⊙O is O. Find the measure of ∠BCO. | 6975.png | [
"Shape(OCA,AC)",
"Shape(OAB,BA)",
"Shape(CA,AB,BO,OC)",
"Shape(CO,OB,BC)",
"Shape(OBC,CB)",
"Cocircular(O,ABC)"
] | [
"Equal(MeasureOfAngle(CAB),64)",
"IsCentreOfCircle(O,O)"
] | [] | Value(MeasureOfAngle(BCO)) | 26 | [
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_center_angle(1,OBC,O)",
"radius_of_circle_property_length_equal(1,OC,O)",
"radius_of_circle_property_length_equal(1,OB,O)",
"isosceles_triangle_judgment_line_equal(1,OBC)",
"isosceles_triangle_property_angle_equal(1,OBC)",
"triangle_pro... | {"START": ["arc_property_circumference_angle_external(1,OBC,A)", "arc_property_center_angle(1,OBC,O)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "triangle_property_angle_sum(1,OBC)"], "isosceles_triangle_judgment_line_equal(1,OBC)": ["isosceles_triangle_property... | |
6,976 | ChengQin_2023-06-04 | GeoQA-angle-10230 | 7 | 如图所示，AD=AC，∠BAD=30°，⊙O的圆心为O，圆O的直径为AB。求∠EOC的大小。 | As shown in the diagram, AD=AC, ∠BAD=30°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠EOC. | 6976.png | [
"Shape(ODB,BC,CD)",
"Shape(OAD,DA)",
"Shape(DC,CO,OA,AD)",
"Shape(OBE,EC,CB)",
"Shape(CE,EO,OC)",
"Shape(OE,EA,AO)",
"Shape(OEA,AE)",
"Collinear(BCOA)",
"Collinear(ECD)",
"Cocircular(O,BEAD)"
] | [
"Equal(LengthOfLine(AD),LengthOfLine(AC))",
"Equal(MeasureOfAngle(BAD),30)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(EOC)) | 30 | [
"isosceles_triangle_judgment_line_equal(1,ADC)",
"isosceles_triangle_property_angle_equal(1,ADC)",
"triangle_property_angle_sum(1,ADC)",
"arc_property_circumference_angle_external(1,OEA,D)",
"arc_property_center_angle(1,OEA,O)",
"flat_angle(1,AOC)",
"angle_addition(1,AOE,EOC)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ADC)", "triangle_property_angle_sum(1,ADC)", "arc_property_circumference_angle_external(1,OEA,D)", "arc_property_center_angle(1,OEA,O)", "flat_angle(1,AOC)", "angle_addition(1,AOE,EOC)"], "isosceles_triangle_judgment_line_equal(1,ADC)": ["isosceles_triangle_property_... | |
6,977 | ChengQin_2023-06-04 | GeoQA-angle-10231 | 7 | 如图所示，∠CDA=115°，⊙O的圆心为O，圆O的直径为AB，CD是⊙O的切线。求∠ACD的大小。 | As shown in the diagram, ∠CDA=115°, O is the center of circle O, the diameter of ⊙O is AB, CD is the tangent to ⊙O. Find the measure of ∠ACD. | 6977.png | [
"Shape(ODA,AD)",
"Shape(DA,AO,OD)",
"Shape(DO,OB,OBD)",
"Shape(OBD,BC,CD)",
"Shape(OA,OAB,BO)",
"Collinear(AOBC)",
"Cocircular(O,ABD)"
] | [
"Equal(MeasureOfAngle(CDA),115)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(CD,O)"
] | [] | Value(MeasureOfAngle(ACD)) | 40 | [
"tangent_of_circle_property_perpendicular(2,CD,O,O)",
"angle_addition(1,CDO,ODA)",
"radius_of_circle_property_length_equal(1,OA,O)",
"radius_of_circle_property_length_equal(1,OD,O)",
"isosceles_triangle_judgment_line_equal(1,ODA)",
"isosceles_triangle_property_angle_equal(1,ODA)",
"triangle_property_ang... | {"START": ["tangent_of_circle_property_perpendicular(2,CD,O,O)", "angle_addition(1,CDO,ODA)", "radius_of_circle_property_length_equal(1,OA,O)", "radius_of_circle_property_length_equal(1,OD,O)", "triangle_property_angle_sum(1,DAC)"], "isosceles_triangle_judgment_line_equal(1,ODA)": ["isosceles_triangle_property_angle_eq... | |
6,978 | ChengQin_2023-06-04 | GeoQA-angle-10232 | 6 | 如图所示，∠BTA=40°，⊙O的圆心为O，⊙O的直径为AB，TA是⊙O的切线。求∠CDB的大小。 | As shown in the diagram, ∠BTA=40°, O is the center of ⊙O, AB is the diameter of circle O, TA is the tangent to ⊙O. Find the measure of ∠CDB. | 6978.png | [
"Shape(OBC,CB)",
"Shape(BC,CE,EB)",
"Shape(ODB,BD)",
"Shape(BE,ED,DB)",
"Shape(EC,CA,AO,OE)",
"Shape(DE,EO,OA,OAD)",
"Shape(OCA,AC)",
"Shape(CT,TA,OCA)",
"Collinear(CED)",
"Collinear(BEOA)",
"Collinear(TCB)",
"Cocircular(O,ADBC)"
] | [
"Equal(MeasureOfAngle(BTA),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(TA,O)"
] | [] | Value(MeasureOfAngle(CDB)) | 40 | [
"tangent_of_circle_property_perpendicular(2,TA,O,O)",
"triangle_property_angle_sum(1,BTA)",
"diameter_of_circle_property_right_angle(1,BCA,O)",
"triangle_property_angle_sum(1,ABC)",
"arc_property_circumference_angle_external(1,OBC,A)",
"arc_property_circumference_angle_external(1,OBC,D)"
] | {"START": ["tangent_of_circle_property_perpendicular(2,TA,O,O)", "triangle_property_angle_sum(1,BTA)", "diameter_of_circle_property_right_angle(1,BCA,O)", "triangle_property_angle_sum(1,ABC)", "arc_property_circumference_angle_external(1,OBC,A)", "arc_property_circumference_angle_external(1,OBC,D)"]} | |
6,979 | ChengQin_2023-06-04 | GeoQA-angle-10233 | 3 | 如图所示，∠FOA=40°，⊙O的圆心为O，圆O的直径为AB。求∠BEF的大小。 | As shown in the diagram, ∠FOA=40°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠BEF. | 6979.png | [
"Shape(OEA,AC,CE)",
"Shape(OBE,EB)",
"Shape(EC,CO,OB,BE)",
"Shape(OAF,FC,CA)",
"Shape(CF,FO,OC)",
"Shape(OFB,BO,OF)",
"Collinear(ACOB)",
"Collinear(FCE)",
"Cocircular(O,AFBE)"
] | [
"Equal(MeasureOfAngle(FOA),40)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)"
] | [] | Value(MeasureOfAngle(BEF)) | 70 | [
"adjacent_complementary_angle(1,BOF,FOA)",
"arc_property_center_angle(1,OFB,O)",
"arc_property_circumference_angle_external(1,OFB,E)"
] | {"START": ["adjacent_complementary_angle(1,BOF,FOA)", "arc_property_center_angle(1,OFB,O)", "arc_property_circumference_angle_external(1,OFB,E)"]} | |
6,980 | ChengQin_2023-06-04 | GeoQA-angle-10234 | 11 | 如图所示，∠CBO=70°，O是⊙O的圆心，AB是⊙O的直径，PA是圆O的切线，PO平行于CB。求∠OPC的大小。 | As shown in the diagram, ∠CBO=70°, O is the center of ⊙O, AB is the diameter of circle O, PA is the tangent to circle O, PO∥CB. Find the measure of ∠OPC. | 6980.png | [
"Shape(AP,PD,OAD)",
"Shape(OAD,DO,OA)",
"Shape(DP,PC,ODC)",
"Shape(ODC,CO,OD)",
"Shape(OC,CB,BO)",
"Shape(OCB,BC)",
"Shape(OB,OBA,AO)",
"Collinear(AOB)",
"Collinear(PDO)",
"Cocircular(O,ADCB)"
] | [
"Equal(MeasureOfAngle(CBO),70)",
"IsCentreOfCircle(O,O)",
"IsDiameterOfCircle(AB,O)",
"IsTangentOfCircle(PA,O)",
"ParallelBetweenLine(PO,CB)"
] | [] | Value(MeasureOfAngle(OPC)) | 20 | [
"tangent_of_circle_property_perpendicular(1,PA,O,O)",
"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)",
"parallel_property_corresponding_angle(2,BC,OP,A)",
... | {"START": ["tangent_of_circle_property_perpendicular(1,PA,O,O)", "radius_of_circle_property_length_equal(1,OC,O)", "radius_of_circle_property_length_equal(1,OB,O)", "parallel_property_corresponding_angle(2,BC,OP,A)", "parallel_property_alternate_interior_angle(2,PO,CB)", "triangle_property_angle_sum(1,APO)", "radius_of... | |
6,981 | ChengQin_2023-06-04 | GeoQA-angle-10235 | 11 | 如图所示，AB=AC，∠BCA=70°，⊙O的圆心为O，三角形ABC的外心为O。求∠ABD的大小。 | As shown in the diagram, AB=AC, ∠BCA=70°, the center of ⊙O is O, O is the center of the circumscribed circle of triangle ABC. Find the measure of ∠ABD. | 6981.png | [
"Shape(OAB,BA)",
"Shape(AB,BO,OA)",
"Shape(AO,OD,DA)",
"Shape(OB,BE,EO)",
"Shape(DO,OC,CD)",
"Shape(OE,EC,CO)",
"Shape(OBC,CB)",
"Shape(OCA,AD,DC)",
"Collinear(ADC)",
"Collinear(BOD)",
"Collinear(AOE)",
"Collinear(BEC)",
"Cocircular(O,ABC)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AC))",
"Equal(MeasureOfAngle(BCA),70)",
"IsCentreOfCircle(O,O)",
"IsCircumcenterOfTriangle(O,ABC)"
] | [] | Value(MeasureOfAngle(ABD)) | 20 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,ABC)",
"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)",
"mirror_congruent... | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,ABC)", "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)", "angle_addition(1,CAO,OAB)"], "isosceles_triangle_judgment_line_equal(1,... |
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