SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BEC=25°, the center of ⊙O is O, AB is the diameter of ⊙O, CH⊥AH. Find the measure of ∠DAB.
问题：如图所示，∠BEC=25°，圆O的圆心为O，⊙O的直径为AB，CH⊥AH。求∠DAB的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CM,MO,OA)
Shape(MC,CH,HM)
Shape(MH,HN,NQ,QP,PM)
Shape(AP,PE,OEA)
Shape(EP,PQ,QE)
Shape(EQ,QD,ODE)
Shape(AO,OM,MP,PA)
Shape(DN,NB,OBD)
Shape(HB,BN,NH)
Shape(BH,HC,OCB)
Shape(ND,DQ,QN)
Collinear(AOMHB)
Collinear(CMPE)
Collinear(CHND)
Collinear(BNQE)
Collinear(APQD)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(MeasureOfAngle(BEC),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CH,AH)
Goal: Value(MeasureOfAngle(DAB))
Answer: 25
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OCB,E)
circle_property_chord_perpendicular_bisect_arc(1,OCD,OHB)
congruent_arc_judgment_length_equal(1,OCB,OBD)
congruent_arc_property_measure_equal(1,OCB,OBD)
Problem: As shown in the diagram, ∠BDC=62°, CE is perpendicular to AE. Find the measure of ∠ACD.
问题：如图所示，∠BDC=62°，CE⊥AE。求∠ACD的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CE,EA)
Shape(DE,EB,BD)
Shape(AE,ED,ODA)
Shape(DB,OBD)
Shape(BE,EC,OCB)
Collinear(CED)
Collinear(AEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),62)
PerpendicularBetweenLine(CE,AE)
Goal: Value(MeasureOfAngle(ACD))
Answer: 28
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
triangle_property_angle_sum(1,ACE)
Problem: As shown in the diagram, EG=GF, ∠DOE=50°, O is the center of circle O, CD is the diameter of ⊙O. Find the measure of ∠FCD.
问题：如图所示，EG=GF，∠DOE=50°，⊙O的圆心为O，CD是圆O的直径。求∠FCD的大小。
Construction (CDL):
Shape(EO,OC,OCE)
Shape(OE,EG,GO)
Shape(CO,OG,GF,FC)
Shape(CF,OFC)
Shape(DG,GE,OED)
Shape(FG,GD,ODF)
Collinear(COGD)
Collinear(EGF)
Cocircular(O,CEDF)
Conditions (CDL):
Equal(LengthOfLine(EG),LengthOfLine(GF))
Equal(MeasureOfAngle(DOE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(FCD))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OED,O)
arc_property_circumference_angle_external(1,ODF,C)
circle_property_chord_perpendicular_bisect_chord(2,O,OG,EF)
circle_property_chord_perpendicular_bisect_arc(1,OEF,OGD)
congruent_arc_judgment_length_equal(1,OED,ODF)
congruent_arc_property_measure_equal(1,OED,ODF)
Problem: As shown in the diagram, ∠CAB=20°, O is the center of ⊙O, the diameter of circle O is AB, DE is perpendicular to AE. Find the measure of ∠DOA.
问题：如图所示，∠CAB=20°，O是圆O的圆心，圆O的直径为AB，DE垂直于AE。求∠DOA的大小。
Construction (CDL):
Shape(CA,AO,OE,EC)
Shape(AC,OCA)
Shape(DO,OA,OAD)
Shape(CE,EB,OBC)
Shape(BE,ED,ODB)
Shape(DO,DE,EO)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(DE,AE)
Goal: Value(MeasureOfAngle(DOA))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,OBC,A)
circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)
congruent_arc_judgment_length_equal(1,ODB,OBC)
congruent_arc_property_measure_equal(1,ODB,OBC)
adjacent_complementary_angle(1,EOD,DOA)
Problem: As shown in the diagram, ∠BCA=15°, O is the center of circle O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=15°，O是圆O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AO,OD,DC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 75
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,OAB)
Problem: As shown in the diagram, ∠DBA=53°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠BCD.
问题：如图所示，∠DBA=53°，O是圆O的圆心，AB是圆O的直径。求∠BCD的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BO,OE,ED)
Shape(DE,EA,OAD)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(BC,CE,EO,OB)
Shape(CB,OBC)
Collinear(BOEA)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),53)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 37
Theorem sequence:
angle_addition(1,BCE,ECA)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
diameter_of_circle_property_right_angle(1,BCA,O)
Problem: As shown in the diagram, ∠BCA=30°, the radius of ⊙O is 1, O is the center of circle O. Find the length of line AB.
问题：如图所示，∠BCA=30°，⊙O的半径为1，O是圆O的圆心。求直线AB的长度。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AO,OD,DC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 1
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)
cosine_theorem(1,OAB)
Problem: As shown in the diagram, ∠CPA=28°, the center of ⊙O is O, the diameter of circle O is AB, AO is perpendicular to CO. Find the measure of ∠ABQ.
问题：如图所示，∠CPA=28°，⊙O的圆心为O，AB是⊙O的直径，AO垂直于CO。求∠ABQ的大小。
Construction (CDL):
Shape(QC,OCQ)
Shape(CQ,QD,DC)
Shape(CD,DB,OBC)
Shape(DO,OB,BD)
Shape(OD,DQ,QO)
Shape(AO,OQ,OQA)
Shape(QP,PA,OQA)
Shape(BO,OA,OAB)
Collinear(PAOB)
Collinear(PQC)
Collinear(QDB)
Collinear(CDO)
Cocircular(O,ABCQ)
Conditions (CDL):
Equal(MeasureOfAngle(CPA),28)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AO,CO)
Goal: Value(MeasureOfAngle(ABQ))
Answer: 17
Theorem sequence:
angle_addition(1,AOQ,QOD)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OQ,O)
arc_property_center_angle(1,OQA,O)
arc_property_circumference_angle_external(1,OQA,B)
isosceles_triangle_judgment_line_equal(1,OCQ)
isosceles_triangle_property_angle_equal(1,OCQ)
triangle_property_angle_sum(1,CQO)
triangle_property_angle_sum(1,CPO)
Problem: As shown in the diagram, ∠OCD=40°, the center of ⊙O is O, AB is the diameter of circle O, CE is perpendicular to BE. Find the measure of ∠DBA.
问题：如图所示，∠OCD=40°，O是圆O的圆心，AB是圆O的直径，CE⊥BE。求∠DBA的大小。
Construction (CDL):
Shape(CE,EO,OC)
Shape(AE,EC,OCA)
Shape(CE,EA,OAB)
Shape(BO,OE,ED,DB)
Shape(CO,OB,OBC)
Shape(BD,ODB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(OCD),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(DBA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OAD,B)
circle_property_chord_perpendicular_bisect_arc(1,OCD,OEA)
congruent_arc_judgment_length_equal(1,OAD,OCA)
congruent_arc_property_measure_equal(1,OAD,OCA)
triangle_property_angle_sum(1,CEO)
Problem: As shown in the diagram, ∠ABC=50°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CDB.
问题：如图所示，∠ABC=50°，O是⊙O的圆心，AB是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(ED,DB,BE)
Shape(BD,ODB)
Shape(DE,EO,OA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,CAB)
diameter_of_circle_property_right_angle(1,BCA,O)
Problem: As shown in the diagram, ∠COA=70°, O is the center of circle O, AB is the diameter of circle O, CD and BA are opposite sides of the ▱ DCBA. Find the measure of ∠BAD.
问题：如图所示，∠COA=70°，O是⊙O的圆心，⊙O的直径为AB，DA和CB是平行四边形DCBA的一组对边。求∠BAD的大小。
Construction (CDL):
Shape(AD,DC,OAC)
Shape(CO,OA,OAC)
Shape(OC,CB,BO)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Parallelogram(DCBA)
Goal: Value(MeasureOfAngle(BAD))
Answer: 145
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,DCBA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
parallel_property_ipsilateral_internal_angle(1,AB,DC)
Problem: As shown in the diagram, ∠ABD=57°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=57°，O是⊙O的圆心，圆O的直径为AB。求∠DCB的大小。
Construction (CDL):
Shape(AO,OE,ED,DA)
Shape(AD,ODA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(BE,EC,OCB)
Shape(BC,OCB)
Shape(CE,EO,OA,OAC)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),57)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 33
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠COB=50°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠CDB.
问题：如图所示，∠COB=50°，O是⊙O的圆心，AB是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(EC,CO,OE)
Shape(BD,ODB)
Shape(CE,EB,OBC)
Shape(AO,OC,OCA)
Shape(DE,EO,OA,OAD)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠ABC=54°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠CDB.
问题：如图所示，∠ABC=54°，O是圆O的圆心，AB是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(ED,DB,BE)
Shape(BD,ODB)
Shape(DE,EO,OA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),54)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 36
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BAC=40°, O is the center of circle O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=40°，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(BO,OC,CB)
Shape(BC,OCB)
Shape(OB,BA,AC,CO)
Shape(AB,OBA)
Shape(CA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, the length of ⌒OAB is equal to the length of ⌒OCA, ∠CAB=50°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠AEC.
问题：如图所示，⌒OAB的长度与⌒OCA的长度相等，∠CAB=50°，O是圆O的圆心，圆O的直径为AB。求∠AEC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(DE,EC,CD)
Shape(ED,DB,OBE)
Shape(CE,OEC)
Collinear(BDC)
Collinear(ADE)
Cocircular(O,ABEC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OCA))
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AEC))
Answer: 65
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OCA)
congruent_arc_property_measure_equal(1,OAB,OCA)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,E)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠COA=110°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CBA.
问题：如图所示，∠COA=110°，O是⊙O的圆心，AB是⊙O的直径。求∠CBA的大小。
Construction (CDL):
Shape(BA,AD,DB)
Shape(AB,OBA)
Shape(OD,DC,CO)
Shape(CD,DA,OAC)
Shape(BD,DO,OC,OCB)
Collinear(BDC)
Collinear(ADO)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠COB=80°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=80°，⊙O的圆心为O，AB是圆O的直径。求∠CAB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(BA,OAB)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOA=60°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠BCA.
问题：如图所示，∠BOA=60°，O是圆O的圆心，圆O的直径为AB。求∠BCA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AO,OD,DC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠COA=110°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CBA.
问题：如图所示，∠COA=110°，O是圆O的圆心，AB是圆O的直径。求∠CBA的大小。
Construction (CDL):
Shape(BA,AO,OC,CB)
Shape(AB,OBA)
Shape(BC,OCB)
Shape(CO,OA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠CBO=25°, O is the center of ⊙O, AO⊥BO. Find the measure of ∠DAC.
问题：如图所示，∠CBO=25°，O是⊙O的圆心，AO垂直于BO。求∠DAC的大小。
Construction (CDL):
Shape(AC,CD,DO,OA)
Shape(CA,OAC)
Shape(BD,DC,OCB)
Shape(OD,DB,BO)
Shape(AO,OB,OBA)
Collinear(CDB)
Collinear(AOD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AO,BO)
Goal: Value(MeasureOfAngle(DAC))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
triangle_property_angle_sum(1,ACD)
triangle_property_angle_sum(1,ODB)
adjacent_complementary_angle(1,CDO,ODB)
adjacent_complementary_angle(1,AOB,BOD)
Problem: As shown in the diagram, ∠CAB=42°, ∠DBC=Mul(2,MeasureOfAngle(CDB))°, the center of ⊙A is A. Find the measure of ∠DAC.
问题：如图所示，∠CAB=42°，∠DBC=Mul(2,MeasureOfAngle(CDB))°，A是圆A的圆心。求∠DAC的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,ED,DA)
Shape(BC,CE,EB)
Shape(CD,DE,EC)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(BA,AD,ODB)
Collinear(AEC)
Collinear(BED)
Cocircular(A,BCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),42)
Equal(MeasureOfAngle(DBC),Mul(2,MeasureOfAngle(CDB)))
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfAngle(DAC))
Answer: 84
Theorem sequence:
arc_property_center_angle(1,ABC,A)
arc_property_center_angle(1,ACD,A)
arc_property_circumference_angle_external(1,ABC,D)
arc_property_circumference_angle_external(1,ACD,B)
Problem: As shown in the diagram, ∠BPA=15°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BPA=15°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OA,AC,CO)
Shape(PC,CB,BP)
Shape(BC,CA,OAB)
Shape(PB,OBP)
Shape(AO,OC,CP,OPA)
Collinear(ACP)
Collinear(OCB)
Cocircular(O,ABP)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,P)
Problem: As shown in the diagram, ∠BAC=60°, O is the center of ⊙O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=60°，O是圆O的圆心。求∠BOC的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AC,CO,OB)
Shape(CA,OAC)
Shape(BO,OC,OCB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠ABE=50°, O is the center of circle O, AB is the diameter of ⊙O, DN⊥AN. Find the measure of ∠DCA.
问题：如图所示，∠ABE=50°，O是圆O的圆心，AB是圆O的直径，DN⊥AN。求∠DCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AM,MC)
Shape(MD,DN,NO,OM)
Shape(EN,NB,BE)
Shape(EB,OBE)
Shape(BN,ND,DB)
Shape(BD,ODB)
Shape(DM,MA,OAD)
Shape(CM,MO,ON,NE,OEC)
Collinear(AMONB)
Collinear(CMD)
Collinear(END)
Cocircular(O,ADBEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(DN,AN)
Goal: Value(MeasureOfAngle(DCA))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
circle_property_chord_perpendicular_bisect_chord(1,O,ON,DE)
circle_property_chord_perpendicular_bisect_arc(1,ODE,ONB)
congruent_arc_judgment_length_equal(1,ODB,OBE)
congruent_arc_property_chord_equal(1,ODB,OBE)
isosceles_triangle_judgment_line_equal(1,BED)
isosceles_triangle_property_angle_equal(1,BED)
mirror_congruent_triangle_judgment_sas(1,ENB,DBN)
mirror_congruent_triangle_property_angle_equal(1,BEN,BND)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CAB=40°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOC=50°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠BAC.
问题：如图所示，∠BOC=50°，O是圆O的圆心，AB是⊙O的直径。求∠BAC的大小。
Construction (CDL):
Shape(AC,CO,OA)
Shape(CA,OAC)
Shape(AO,OB,OBA)
Shape(BO,OC,OCB)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠DAB=60°, the center of ⊙O is O, the diameter of ⊙O is AB, AD∥OC. Find the measure of ∠DAC.
问题：如图所示，∠DAB=60°，O是圆O的圆心，AB是⊙O的直径，AD∥OC。求∠DAC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AC,OCD)
Shape(CO,OB,OBC)
Shape(AO,OC,CA)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,OC)
Goal: Value(MeasureOfAngle(DAC))
Answer: 30
Theorem sequence:
parallel_property_corresponding_angle(2,AD,OC,B)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
parallel_property_alternate_interior_angle(1,AD,OC)
Problem: As shown in the diagram, ∠ABC=30°, C is the center of ⊙C, the diameter of circle C is EF, BC⊥AC. Find the measure of ∠BED.
问题：如图所示，∠ABC=30°，圆C的圆心为C，圆C的直径为EF，BC⊥AC。求∠BED的大小。
Construction (CDL):
Shape(DB,BF,ODF)
Shape(FC,CG,GD,ODF)
Shape(DA,OAD)
Shape(AD,DG,GA)
Shape(GC,CE,EG)
Shape(AG,GE,OEA)
Shape(EC,CF,OFE)
Collinear(BDA)
Collinear(BFCE)
Collinear(AGC)
Collinear(DGE)
Cocircular(C,ADFE)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
IsCentreOfCircle(C,C)
IsDiameterOfCircle(EF,C)
PerpendicularBetweenLine(BC,AC)
Goal: Value(MeasureOfAngle(BED))
Answer: 15
Theorem sequence:
arc_property_center_angle(1,CEA,C)
arc_property_circumference_angle_external(1,CEA,D)
adjacent_complementary_angle(1,ADG,GDB)
adjacent_complementary_angle(1,BCG,GCE)
triangle_property_angle_sum(1,DBE)
Problem: As shown in the diagram, ∠OAB=40°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠BCA.
问题：如图所示，∠OAB=40°，O是圆O的圆心，⊙O的直径为AB。求∠BCA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(BC,CD,DB)
Shape(BD,DA,AB)
Shape(AD,DO,OA)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AO,OD,DC,OCA)
Collinear(CDA)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,ABO)
Problem: As shown in the diagram, ∠COB=100°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=100°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BC,CO)
Shape(CB,OBC)
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, ∠BOA=140°, the center of ⊙O is O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=140°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CA,OAC)
Shape(CB,BO,OA,AC)
Shape(AO,OB,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=25°, ∠DEC=30°, the center of circle O is O. Find the measure of ∠DOB.
问题：如图所示，∠CAB=25°，∠DEC=30°，O是⊙O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BF,FA)
Shape(AF,FE,OEA)
Shape(FB,BM,MF)
Shape(CM,MB,OBC)
Shape(OM,MC,CN,NO)
Shape(DN,NC,OCD)
Shape(EN,ND,DE)
Shape(ED,ODE)
Shape(EF,FM,MO,ON,NE)
Collinear(AFMC)
Collinear(BMO)
Collinear(DNO)
Collinear(ENC)
Collinear(BFE)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
Equal(MeasureOfAngle(DEC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 110
Theorem sequence:
angle_addition(1,DEN,NEF)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,E)
arc_property_circumference_angle_external(1,OBD,E)
Problem: As shown in the diagram, OA=1, ∠AOB=90°, ∠BCO=60°, the center of circle P is P. Find the length of line OB.
问题：如图所示，OA=1，∠AOB=90°，∠BCO=60°，P是⊙P的圆心。求直线OB的长度。
Construction (CDL):
Shape(OA,PAO)
Shape(AD,DC,PCA)
Shape(CD,DB,BC)
Shape(CB,PBC)
Shape(BO,POB)
Shape(AO,OD,DA)
Shape(DO,OB,BD)
Collinear(ADB)
Collinear(CDO)
Cocircular(P,AOBC)
Conditions (CDL):
Equal(LengthOfLine(OA),1)
Equal(MeasureOfAngle(AOB),90)
Equal(MeasureOfAngle(BCO),60)
IsCentreOfCircle(P,P)
Goal: Value(LengthOfLine(OB))
Answer: sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,POB,A)
arc_property_circumference_angle_external(1,POB,C)
triangle_property_angle_sum(1,AOB)
sine_theorem(1,OBA)
Problem: As shown in the diagram, ∠CAD=40°, O is the center of ⊙O, BG⊥OG. Find the measure of ∠DOB.
问题：如图所示，∠CAD=40°，圆O的圆心为O，BG垂直于OG。求∠DOB的大小。
Construction (CDL):
Shape(AE,EO,OC,CA)
Shape(AC,OCA)
Shape(CO,OG,GC)
Shape(OE,EF,FG,GO)
Shape(EB,BF,FE)
Shape(BE,EA,OAB)
Shape(DF,FB,OBD)
Shape(FD,DG,GF)
Shape(CG,GD,ODC)
Collinear(AEFD)
Collinear(BEO)
Collinear(BFGC)
Collinear(OGD)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),40)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BG,OG)
Goal: Value(MeasureOfAngle(DOB))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,ODC,A)
circle_property_chord_perpendicular_bisect_arc(1,OBC,OGD)
congruent_arc_judgment_length_equal(1,OBD,ODC)
congruent_arc_property_measure_equal(1,OBD,ODC)