SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ABO=20°, ∠CDA=15°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠ABO=20°，∠CDA=15°，O是圆O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(DO,OB,OBD)
Shape(AO,OD,DA)
Shape(OC,CB,BO)
Shape(AC,CO,OA)
Collinear(DOC)
Collinear(ACB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),20)
Equal(MeasureOfAngle(CDA),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 35
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,OAB)
angle_addition(1,DAO,OAC)
Problem: As shown in the diagram, ∠ADC=32°, the center of ⊙O is O, the diameter of ⊙O is BC. Find the measure of ∠CAO.
问题：如图所示，∠ADC=32°，⊙O的圆心为O，BC是圆O的直径。求∠CAO的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(DE,EB,OBD)
Shape(CD,ODC)
Shape(AC,OCA)
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(AE,EC,CA)
Shape(ED,DC,CE)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 58
Theorem sequence:
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)
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)
angle_addition(1,CAO,OAB)
Problem: As shown in the diagram, ∠CDO=35°, the center of circle O is O, the diameter of ⊙O is AB, BC∥DO. Find the measure of ∠OAC.
问题：如图所示，∠CDO=35°，圆O的圆心为O，圆O的直径为AB，BC平行于DO。求∠OAC的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CA,OAC)
Shape(DE,EB,OBD)
Shape(AO,OD,ODA)
Shape(AC,CO,OA)
Shape(OC,CE,EO)
Shape(EC,CB,BE)
Shape(OE,ED,DO)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(CDO),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(BC,DO)
Goal: Value(MeasureOfAngle(OAC))
Answer: 20
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OCD)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OCD)
parallel_property_alternate_interior_angle(2,BC,DO)
angle_addition(1,ACO,OCB)
angle_addition(1,OCE,ECB)
Problem: As shown in the diagram, BC=3, ∠BDC=30°, O is the center of ⊙O, the diameter of circle O is AB. Find the length of line AB.
问题：如图所示，BC=3，∠BDC=30°，圆O的圆心为O，圆O的直径为AB。求直线AB的长度。
Construction (CDL):
Shape(CA,OAC)
Shape(BC,OCB)
Shape(DB,OBD)
Shape(AC,CE,EA)
Shape(EC,CB,BE)
Shape(DE,EB,BD)
Shape(AE,ED,ODA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(BDC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(AB))
Answer: 6
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
diameter_of_circle_property_right_angle(1,ACB,O)
sine_theorem(1,BAC)
Problem: As shown in the diagram, ∠CDA=40°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠BAC.
问题：如图所示，∠CDA=40°，O是圆O的圆心，AB是⊙O的直径。求∠BAC的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CA,OAC)
Shape(AD,ODA)
Shape(DE,EB,OBD)
Shape(BE,EC,CB)
Shape(ED,DA,AE)
Shape(CE,EA,AC)
Collinear(BOEA)
Collinear(CED)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, ∠BOA=120°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=120°，O是⊙O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(OA,AC,CB,BO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 120
Theorem sequence:
round_angle(1,AOB,BOA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BCA=30°, O is the center of ⊙O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=30°，⊙O的圆心为O。求∠OAB的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(OA,AE,EO)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 60
Theorem sequence:
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_property_angle_sum(1,ABO)
Problem: As shown in the diagram, AD=AC, ∠BOC=112°, O is the center of ⊙O. Find the measure of ∠BDC.
问题：如图所示，AD=AC，∠BOC=112°，O是⊙O的圆心。求∠BDC的大小。
Construction (CDL):
Shape(BO,OC,OCB)
Shape(AB,OBA)
Shape(CE,OEC)
Shape(EC,CA,OAE)
Shape(AC,CO,OB,BA)
Shape(AD,DE,OAE)
Collinear(CED)
Collinear(BAD)
Cocircular(O,BAEC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(AC))
Equal(MeasureOfAngle(BOC),112)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 28
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
adjacent_complementary_angle(1,BAC,CAD)
isosceles_triangle_judgment_line_equal(1,ADC)
isosceles_triangle_property_angle_equal(1,ADC)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, ∠COB=40°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=40°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(AE,EO,OC,OCA)
Shape(EA,AB,BE)
Shape(OE,EC,CO)
Shape(CE,EB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(OEB)
Collinear(CEA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ABD=20°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BCD.
问题：如图所示，∠ABD=20°，⊙O的圆心为O，圆O的直径为AB。求∠BCD的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 110
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, CA=CD, ∠DCA=40°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CAB.
问题：如图所示，CA=CD，∠DCA=40°，圆O的圆心为O，AB是圆O的直径。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DA,OAD)
Shape(CB,OBC)
Shape(BE,ED,ODB)
Shape(AE,EC,CA)
Shape(CE,EB,BC)
Shape(AD,DE,EA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(CA),LengthOfLine(CD))
Equal(MeasureOfAngle(DCA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 20
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CAD)
isosceles_triangle_property_angle_equal(1,CAD)
triangle_property_angle_sum(1,ADC)
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,ABC)
Problem: As shown in the diagram, ∠ABC=66°, the center of circle O is O, AB is the diameter of ⊙O, AF is perpendicular to EF. Find the measure of ∠FEA.
问题：如图所示，∠ABC=66°，O是⊙O的圆心，圆O的直径为AB，AF垂直于EF。求∠FEA的大小。
Construction (CDL):
Shape(AE,EC,OCA)
Shape(EA,AO,OF,FE)
Shape(EF,FB,BC,CE)
Shape(CB,OBC)
Shape(BF,FO,OA,OAB)
Collinear(AOFB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),66)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AF,EF)
Goal: Value(MeasureOfAngle(FEA))
Answer: 66
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,EAF)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BCA=38°, the center of ⊙O is O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=38°，O是圆O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AE,EO,OA)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),38)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 52
Theorem sequence:
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_property_angle_sum(1,ABO)
Problem: As shown in the diagram, ∠BCA=60°, the center of circle O is O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=60°，圆O的圆心为O。求∠OAB的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OB,BC,CA)
Shape(OAB,BA)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 30
Theorem sequence:
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)
sine_theorem(1,OAB)
Problem: As shown in the diagram, ∠ABC=50°, the center of ⊙O is O, D bisects arc OCA. Find the measure of ∠DAB.
问题：如图所示，∠ABC=50°，圆O的圆心为O，D是⌒OCA的中点。求∠DAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OBC,CB)
Shape(AO,OB,BC,OCD,DA)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsCentreOfCircle(O,O)
IsMidpointOfArc(D,OCA)
Goal: Value(MeasureOfAngle(DAB))
Answer: 65
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OBD,A)
congruent_arc_judgment_length_equal(1,OCD,ODA)
congruent_arc_property_measure_equal(1,OCD,ODA)
arc_addition_measure(1,OCD,ODA)
flat_angle(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_addition_measure(1,OBD,ODA)
Problem: As shown in the diagram, ∠BCD=40°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DBA.
问题：如图所示，∠BCD=40°，O是⊙O的圆心，AB是圆O的直径。求∠DBA的大小。
Construction (CDL):
Shape(OA,AD,DM,MO)
Shape(MD,DB,BM)
Shape(CM,MB,BC)
Shape(OCA,AO,OM,MC)
Shape(CB,OBC)
Shape(BD,ODB)
Shape(DA,OAD)
Collinear(AOMB)
Collinear(CMD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,ODB,C)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
Problem: As shown in the diagram, ∠CBA=50°, the center of ⊙O is O, AB is the diameter of circle O, D bisects ⌒OAC. Find the measure of ∠BAD.
问题：如图所示，∠CBA=50°，O是⊙O的圆心，圆O的直径为AB，D是⌒OAC的中点。求∠BAD的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(ODC,CD)
Shape(OAD,DA)
Shape(AD,DC,CB,BO,OA)
Collinear(AOB)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(D,OAC)
Goal: Value(MeasureOfAngle(BAD))
Answer: 65
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,OAC,B)
congruent_arc_judgment_length_equal(1,OAD,ODC)
congruent_arc_property_measure_equal(1,OAD,ODC)
flat_angle(1,BOA)
arc_property_center_angle(1,OAB,O)
arc_addition_measure(1,OAD,ODC)
arc_addition_measure(1,OAD,ODB)
Problem: As shown in the diagram, ∠COB=100°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=100°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,OBC,CO)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠EAD=22°, O is the center of circle O, AB is the diameter of circle O, AD⊥FD. Find the measure of ∠BAF.
问题：如图所示，∠EAD=22°，O是⊙O的圆心，⊙O的直径为AB，AD⊥FD。求∠BAF的大小。
Construction (CDL):
Shape(OAM,MA)
Shape(AM,OME,EA)
Shape(OME,MD,DE)
Shape(AE,EF,FA)
Shape(OEF,FE)
Shape(AF,OFB,BO,OA)
Shape(AO,OB,OBA)
Collinear(AMD)
Collinear(AOB)
Collinear(DEF)
Cocircular(O,AMEFB)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),22)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AD,FD)
Goal: Value(MeasureOfAngle(BAF))
Answer: 22
Theorem sequence:
triangle_property_angle_sum(1,ADE)
adjacent_complementary_angle(1,DEA,AEF)
flat_angle(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OFA,E)
arc_property_circumference_angle_external(1,OFB,A)
arc_addition_measure(1,OFB,OBA)
Problem: As shown in the diagram, ∠ABD=55°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DCB.
问题：如图所示，∠ABD=55°，O是⊙O的圆心，AB是圆O的直径。求∠DCB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OM,MD)
Shape(DM,MB,BD)
Shape(DB,OBD)
Shape(MO,OA,OAC,CM)
Shape(BM,MC,CB)
Shape(BC,OCB)
Collinear(DMC)
Collinear(AOMB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, the measure of ⌒OCE is 100, O is the center of ⊙O, the diameter of ⊙O is AB, CE is the diameter of circle O, AB is parallel to CE. Find the measure of ∠COA.
问题：如图所示，弧OCE的角度为100，圆O的圆心为O，AB是圆O的直径，⊙O的直径为CE，AB∥CE。求∠COA的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(OA,OAC,CO)
Shape(DO,OB,OBD)
Shape(BO,OC,CE,OEB)
Shape(EC,OCE)
Collinear(DOC)
Collinear(AOB)
Cocircular(O,ACEBD)
Conditions (CDL):
Equal(MeasureOfArc(OCE),100)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CE,O)
ParallelBetweenLine(AB,CE)
Goal: Value(MeasureOfAngle(COA))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(3,AB,CE,O)
parallel_property_alternate_interior_angle(2,AO,CE)
arc_addition_measure(1,OCE,OED)
flat_angle(1,DOC)
arc_property_center_angle(1,OCD,O)
arc_property_circumference_angle_external(1,OED,C)
Problem: As shown in the diagram, ∠BOA=100°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=100°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=135°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=135°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),135)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 135/2
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BDC=57°, the center of ⊙O is O, CM is perpendicular to AM. Find the measure of ∠ACD.
问题：如图所示，∠BDC=57°，⊙O的圆心为O，CM⊥AM。求∠ACD的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CM,MA)
Shape(O)
Shape(AM,MD,ODA)
Shape(MB,BD,DM)
Shape(DB,OBD)
Shape(MC,OCB,BM)
Collinear(CMD)
Collinear(AMB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),57)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CM,AM)
Goal: Value(MeasureOfAngle(ACD))
Answer: 33
Theorem sequence:
vertical_angle(1,CMA,DMB)
triangle_property_angle_sum(1,MBD)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠DOB=110°, O is the center of ⊙O, the diameter of circle O is AB, AD is parallel to CO. Find the measure of ∠COA.
问题：如图所示，∠DOB=110°，⊙O的圆心为O，AB是⊙O的直径，AD平行于CO。求∠COA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(DO,OB,OBD)
Shape(OA,OAC,CO)
Shape(OC,OCB,BO)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(COA))
Answer: 55
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,CO)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠BAC=36°, ∠OCA=28°, the center of circle O is O. Find the measure of ∠OBA.
问题：如图所示，∠BAC=36°，∠OCA=28°，O是⊙O的圆心。求∠OBA的大小。
Construction (CDL):
Shape(AM,MB,BA)
Shape(AB,OBA)
Shape(MO,OC,CM)
Shape(MC,OCB,BM)
Shape(OM,MA,OAC,CO)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),36)
Equal(MeasureOfAngle(OCA),28)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBA))
Answer: 64
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
triangle_property_angle_sum(1,OCM)
vertical_angle(1,AMB,CMO)
triangle_property_angle_sum(1,AMB)
Problem: As shown in the diagram, ∠CAB=30°, the center of ⊙O is O, CD is the diameter of circle O, AM⊥CM. Find the measure of ∠BCD.
问题：如图所示，∠CAB=30°，O是⊙O的圆心，CD是圆O的直径，AM⊥CM。求∠BCD的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AM,MC)
Shape(CM,MB,BC)
Shape(CB,OBC)
Shape(MA,OAD,DO,OM)
Shape(MO,OB,BM)
Shape(BO,OD,ODB)
Collinear(AMB)
Collinear(CMOD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
PerpendicularBetweenLine(AM,CM)
Goal: Value(MeasureOfAngle(BCD))
Answer: 60
Theorem sequence:
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,BCO)
sine_theorem(1,OBC)
Problem: As shown in the diagram, ∠BCA=60°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=60°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BCA=40°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=40°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,OAB,BO)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CBA=32°, the center of circle O is O, AB is the diameter of circle O. Find the measure of ∠BDC.
问题：如图所示，∠CBA=32°，O是圆O的圆心，圆O的直径为AB。求∠BDC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CM,MA)
Shape(OCB,BC)
Shape(CB,BO,OM,MC)
Shape(AM,MD,ODA)
Shape(MO,OB,BD,DM)
Shape(DB,OBD)
Collinear(CMD)
Collinear(AMOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 58
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠DOC=62°, O is the center of ⊙O, AB is the diameter of circle O, AD∥OC. Find the measure of ∠ABD.
问题：如图所示，∠DOC=62°，圆O的圆心为O，AB是⊙O的直径，AD平行于OC。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(DO,OM,MD)
Shape(DM,MC,OCD)
Shape(MO,OB,BM)
Shape(CM,MB,BC)
Shape(CB,OBC)
Collinear(AOB)
Collinear(OMC)
Collinear(DMB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOC),62)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,OC)
Goal: Value(MeasureOfAngle(ABD))
Answer: 28
Theorem sequence:
parallel_property_alternate_interior_angle(2,AD,OC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
sine_theorem(1,ODA)
triangle_property_angle_sum(1,ODA)
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠CAB=40°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=40°，圆O的圆心为O，圆O的直径为AB。求∠ADC的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(AD,DM,MO,OA)
Shape(AO,OM,MC,CA)
Shape(AC,OCA)
Shape(MD,ODB,BO)
Shape(MB,BC,CM)
Shape(CB,OBC)
Collinear(AOMB)
Collinear(DMC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠ABC=28°, the center of circle O is O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=28°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BM,MA)
Shape(AM,MC,CA)
Shape(AC,OCA)
Shape(CM,MO,OC)
Shape(OM,MB,OBC,CO)
Collinear(AMO)
Collinear(BMC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),28)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 56
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠CBA=25°, O is the center of circle O. Find the measure of ∠COA.
问题：如图所示，∠CBA=25°，圆O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(OM,MB,OBA,AO)
Shape(OA,AM,MO)
Shape(BM,MC,CB)
Shape(MA,OAC,CM)
Shape(BC,OCB)
Collinear(OMC)
Collinear(AMB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠ABD=20°, the center of circle O is O, BD is the diameter of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠ABD=20°，圆O的圆心为O，⊙O的直径为BD。求∠BCA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OM,MA)
Shape(AM,MD,ODA)
Shape(MO,OB,BC,CM)
Shape(CB,OBC)
Shape(DM,MC,OCD)
Collinear(BOMD)
Collinear(AMC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 70
Theorem sequence:
flat_angle(1,BOD)
arc_property_center_angle(1,ODB,O)
arc_addition_measure(1,ODA,OAB)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠BPA=40°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BPA=40°，O是⊙O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OPA,AP)
Shape(PA,AO,OB,BP)
Shape(OA,OAB,BO)
Shape(PB,OBP)
Cocircular(O,ABP)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,P)