SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=CD, ∠BDC=44°, ∠DBA=100°, E is the midpoint of segment BC, F bisects segment AD, G is the midpoint of segment BD. Find the measure of ∠GEF.
问题：如图所示，AB=CD，∠BDC=44°，∠DBA=100°，E是线段BC的中点，F是线段AD的中点，G平分线段BD。求∠GEF的大小。
Construction (CDL):
Shape(BA,AF,FH,HB)
Shape(BH,HE,EB)
Shape(EH,HG,GE)
Shape(CE,EG,GD,DC)
Shape(GH,HF,FG)
Shape(GF,FD,DG)
Collinear(BEC)
Collinear(BHGD)
Collinear(EHF)
Collinear(AFD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(MeasureOfAngle(BDC),44)
Equal(MeasureOfAngle(DBA),100)
IsMidpointOfLine(E,BC)
IsMidpointOfLine(F,AD)
IsMidpointOfLine(G,BD)
Goal: Value(MeasureOfAngle(GEF))
Answer: 28
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,GF,DBA)
midsegment_of_triangle_judgment_midpoint(1,GE,BDC)
midsegment_of_triangle_property_parallel(1,GF,DBA)
midsegment_of_triangle_property_parallel(1,GE,BDC)
midsegment_of_triangle_property_length(1,GF,DBA)
midsegment_of_triangle_property_length(1,GE,BDC)
parallel_property_corresponding_angle(1,GE,DC,B)
parallel_property_ipsilateral_internal_angle(1,GF,BA)
angle_addition(1,FGH,HGE)
isosceles_triangle_judgment_line_equal(1,GEF)
isosceles_triangle_property_angle_equal(1,GEF)
triangle_property_angle_sum(1,EFG)
Problem: As shown in the diagram, AB=AC, ∠CAB=40°, BA and CD are opposite sides of the parallelogram ABCD. Find the measure of ∠CDA.
问题：如图所示，AB=AC，∠CAB=40°，BA和CD是▱ABCD的一组对边。求∠CDA的大小。
Construction (CDL):
Shape(DA,AC,CD)
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),40)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(CDA))
Answer: 70
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,CAB)
parallelogram_property_opposite_angle_equal(1,BCDA)
Problem: As shown in the diagram, ∠DGJ=90°, ∠EFD=90°, ∠FDE=45°. Find the measure of ∠FEG minus the measure of ∠EFD.
问题：如图所示，∠DGJ=90°，∠EFD=90°，∠FDE=45°。求∠FEG的大小减去∠EFD的大小。
Construction (CDL):
Shape(AB,BH,HG,GE,EA)
Shape(DE,EF,FD)
Shape(FE,EG,GH,HI,IF)
Shape(FI,IC,CF)
Shape(HJ,JI,IH)
Collinear(AEFC)
Collinear(DFIJ)
Collinear(DEG)
Collinear(BHIC)
Conditions (CDL):
Equal(MeasureOfAngle(DGJ),90)
Equal(MeasureOfAngle(EFD),90)
Equal(MeasureOfAngle(FDE),45)
Goal: Value(Sub(MeasureOfAngle(FEG),MeasureOfAngle(EFD)))
Answer: 45
Theorem sequence:
triangle_property_angle_sum(1,DEF)
adjacent_complementary_angle(1,DEF,FEG)
Problem: As shown in the diagram, ∠BAC=30°. Find the sum of the measure of ∠DBC and the measure of ∠BCE.
问题：如图所示，∠BAC=30°。求∠DBC的大小与∠BCE的大小之和。
Construction (CDL):
Shape(BA,AC,CB)
Shape(BC,CE,ED,DB)
Collinear(ABD)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
Goal: Value(Add(MeasureOfAngle(DBC),MeasureOfAngle(BCE)))
Answer: 210
Theorem sequence:
triangle_property_angle_sum(1,BAC)
adjacent_complementary_angle(1,DBC,CBA)
adjacent_complementary_angle(1,ACB,BCE)
Problem: As shown in the diagram, Mul(1/4=MeasureOfAngle(ABC)), Mul(2=MeasureOfAngle(ABC)), AD bisects ∠CAB, AE⊥CE. Find the measure of ∠ECA.
问题：如图所示，Mul(1/4=MeasureOfAngle(ABC))，Mul(2=MeasureOfAngle(ABC))，AD平分∠CAB，AE⊥CE。求∠ECA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(AE,EC,CA)
Collinear(BDEC)
Conditions (CDL):
Equal(Mul(1/4,MeasureOfAngle(ABC)),MeasureOfAngle(EAD))
Equal(Mul(2,MeasureOfAngle(ABC)),MeasureOfAngle(CAB))
IsBisectorOfAngle(AD,CAB)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(ECA))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,AEC)
angle_addition(1,CAD,DAB)
angle_addition(1,CAE,EAD)
Problem: As shown in the diagram, ∠DAE=20°, ∠ECD=35°, ∠FBE=45°. Find the measure of ∠DFE.
问题：如图所示，∠DAE=20°，∠ECD=35°，∠FBE=45°。求∠DFE的大小。
Construction (CDL):
Shape(AF,FD,DA)
Shape(FB,BE,EF)
Shape(DF,FE,EC,CD)
Collinear(AFE)
Collinear(ADC)
Collinear(DFB)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),20)
Equal(MeasureOfAngle(ECD),35)
Equal(MeasureOfAngle(FBE),45)
Goal: Value(MeasureOfAngle(DFE))
Answer: 100
Theorem sequence:
triangle_property_angle_sum(1,FBE)
triangle_property_angle_sum(1,AEC)
adjacent_complementary_angle(1,DFE,EFB)
adjacent_complementary_angle(1,BEF,FEC)
Problem: As shown in the diagram, ∠ABD=30°, ∠BDA=80°, BD bisects ∠ABC. Find the measure of ∠BCD.
问题：如图所示，∠ABD=30°，∠BDA=80°，BD是∠ABC的角平分线。求∠BCD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),30)
Equal(MeasureOfAngle(BDA),80)
IsBisectorOfAngle(BD,ABC)
Goal: Value(MeasureOfAngle(BCD))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,CDB,BDA)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠ABP=20°, ∠ACP=50°, BP is the angle bisector of ∠ABC, CP bisects ∠ACM. Find the measure of ∠CPD.
问题：如图所示，∠ABP=20°，∠ACP=50°，BP是∠ABC的角平分线，CP是∠ACM的角平分线。求∠CPD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(PD,DC,CP)
Shape(BC,CD,DB)
Collinear(ADC)
Collinear(BDP)
Collinear(BCM)
Conditions (CDL):
Equal(MeasureOfAngle(ABP),20)
Equal(MeasureOfAngle(ACP),50)
IsBisectorOfAngle(BP,ABC)
IsBisectorOfAngle(CP,ACM)
Goal: Value(MeasureOfAngle(CPD))
Answer: 30
Theorem sequence:
adjacent_complementary_angle(1,BCP,PCM)
triangle_property_angle_sum(1,CPB)
Problem: As shown in the diagram, ∠CAB=50°, ∠CPB=120°, BD bisects ∠ABP, CE is the angle bisector of ∠PCA. Find the measure of ∠CFB.
问题：如图所示，∠CAB=50°，∠CPB=120°，BD平分∠ABP，CE是∠PCA的角平分线。求∠CFB的大小。
Construction (CDL):
Shape(AE,EF,FD,DA)
Shape(EB,BF,FE)
Shape(DF,FC,CD)
Shape(CF,FB,BP,PC)
Shape(BC,CP,PB)
Collinear(AEB)
Collinear(ADC)
Collinear(EFC)
Collinear(DFB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
Equal(MeasureOfAngle(CPB),120)
IsBisectorOfAngle(BD,ABP)
IsBisectorOfAngle(CE,PCA)
Goal: Value(MeasureOfAngle(CFB))
Answer: 85
Theorem sequence:
triangle_property_angle_sum(1,BCP)
triangle_property_angle_sum(1,CFB)
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABP,PBC)
angle_addition(1,BCP,PCA)
angle_addition(1,EBF,FBP)
angle_addition(1,PCF,FCD)
angle_addition(1,FBP,PBC)
angle_addition(1,BCP,PCF)
Problem: As shown in the diagram, ∠ABC=60°, ∠CAB=55°. Find the measure of ∠BCA.
问题：如图所示，∠ABC=60°，∠CAB=55°。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(CAB),55)
Goal: Value(MeasureOfAngle(BCA))
Answer: 65
Theorem sequence:
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠CAB=60°. Find the sum of the measure of ∠BDE, the measure of ∠DEB, the measure of ∠CFG and the measure of ∠FGC.
问题：如图所示，∠CAB=60°。求∠BDE的大小、∠DEB的大小、∠CFG的大小与∠FGC的大小之和。
Construction (CDL):
Shape(DE,EB,BD)
Shape(AB,BC,CA)
Shape(CF,FG,GC)
Collinear(DBCG)
Collinear(ABE)
Collinear(ACF)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
Goal: Value(Add(MeasureOfAngle(BDE),MeasureOfAngle(DEB),MeasureOfAngle(CFG),MeasureOfAngle(FGC)))
Answer: 240
Theorem sequence:
triangle_property_angle_sum(1,DEB)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,CFG)
adjacent_complementary_angle(1,CBE,EBD)
adjacent_complementary_angle(1,DCA,ACG)
adjacent_complementary_angle(1,ABC,CBE)
adjacent_complementary_angle(1,ACG,GCF)
Problem: As shown in the diagram, ∠ABC=56°, ∠ACD=120°. Find the measure of ∠CAB.
问题：如图所示，∠ABC=56°，∠ACD=120°。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),56)
Equal(MeasureOfAngle(ACD),120)
Goal: Value(MeasureOfAngle(CAB))
Answer: 64
Theorem sequence:
adjacent_complementary_angle(1,BCA,ACD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BAC=∠DBA, ∠CBD=54°, CD bisects ∠ACB, BD is perpendicular to CD. Find the measure of ∠BAC.
问题：如图所示，∠BAC=∠DBA，∠CBD=54°，CD平分∠ACB，BD垂直于CD。求∠BAC的大小。
Construction (CDL):
Shape(BA,AC,CD,DB)
Shape(BD,DC,CB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),MeasureOfAngle(DBA))
Equal(MeasureOfAngle(CBD),54)
IsBisectorOfAngle(CD,ACB)
PerpendicularBetweenLine(BD,CD)
Goal: Value(MeasureOfAngle(BAC))
Answer: 27
Theorem sequence:
triangle_property_angle_sum(1,BDC)
angle_addition(1,ACD,DCB)
angle_addition(1,CBD,DBA)
triangle_property_angle_sum(1,BAC)
Problem: As shown in the diagram, ∠CAB=110°, BO bisects ∠ABC, CO is the angle bisector of ∠BCA. Find the measure of ∠COB.
问题：如图所示，∠CAB=110°，BO平分∠ABC，CO平分∠BCA。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),110)
IsBisectorOfAngle(BO,ABC)
IsBisectorOfAngle(CO,BCA)
Goal: Value(MeasureOfAngle(COB))
Answer: 145
Theorem sequence:
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABO,OBC)
angle_addition(1,BCO,OCA)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠BAC=90°, ∠CBA=45°, ∠CBD=30°. Find the measure of ∠DEA.
问题：如图所示，∠BAC=90°，∠CBA=45°，∠CBD=30°。求∠DEA的大小。
Construction (CDL):
Shape(DC,CE,ED)
Shape(AE,EB,BA)
Shape(EC,CB,BE)
Collinear(DEB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),90)
Equal(MeasureOfAngle(CBA),45)
Equal(MeasureOfAngle(CBD),30)
Goal: Value(MeasureOfAngle(DEA))
Answer: 105
Theorem sequence:
angle_addition(1,CBE,EBA)
triangle_property_angle_sum(1,AEB)
adjacent_complementary_angle(1,DEA,AEB)
Problem: As shown in the diagram, ∠CDE=45°, ∠ECD=90°, ∠EDF=20°, ∠FDG=90°. Find the measure of ∠GDB.
问题：如图所示，∠CDE=45°，∠ECD=90°，∠EDF=20°，∠FDG=90°。求∠GDB的大小。
Construction (CDL):
Shape(EC,CD,DE)
Shape(FD,DG,GF)
Shape(ED,DF,FE)
Collinear(CDB)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),45)
Equal(MeasureOfAngle(ECD),90)
Equal(MeasureOfAngle(EDF),20)
Equal(MeasureOfAngle(FDG),90)
Goal: Value(MeasureOfAngle(GDB))
Answer: 25
Theorem sequence:
angle_addition(1,CDE,EDF)
angle_addition(1,CDF,FDG)
adjacent_complementary_angle(1,CDG,GDB)
Problem: As shown in the diagram, ∠AEO=90°, ∠COA=100°, O is the center of circle O, DB is the diameter of circle O. Find the measure of ∠OCD.
问题：如图所示，∠AEO=90°，∠COA=100°，圆O的圆心为O，⊙O的直径为DB。求∠OCD的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(DO,OC,CD)
Shape(DC,OCD)
Shape(AE,EO,OA)
Shape(OE,EC,CO)
Shape(BE,EA,OAB)
Shape(CE,EB,OBC)
Collinear(DOEB)
Collinear(AEC)
Cocircular(O,DABC)
Conditions (CDL):
Equal(MeasureOfAngle(AEO),90)
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DB,O)
Goal: Value(MeasureOfAngle(OCD))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAC,O)
circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_addition_measure(1,OAB,OBC)
arc_property_circumference_angle_external(1,OBC,D)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCD)
isosceles_triangle_property_angle_equal(1,OCD)
Problem: As shown in the diagram, ∠COA=58°, O is the center of ⊙O. Find the measure of ∠BAO.
问题：如图所示，∠COA=58°，圆O的圆心为O。求∠BAO的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OB)
Shape(CO,OA,OAC)
Shape(BC,OCB)
Collinear(BOC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),58)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAO))
Answer: 29
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
Problem: As shown in the diagram, ∠BAC=40°, O is the center of ⊙O. Find the measure of ∠OCB.
问题：如图所示，∠BAC=40°，⊙O的圆心为O。求∠OCB的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(BO,OC,CB)
Shape(BA,AC,CO,OB)
Shape(AB,OBA)
Shape(CA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCB))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,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)
triangle_property_angle_sum(1,BOC)
Problem: As shown in the diagram, ∠BCD=20°, ∠DBC=35°, the center of circle O is O. Find the measure of ∠OAD.
问题：如图所示，∠BCD=20°，∠DBC=35°，O是圆O的圆心。求∠OAD的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(BD,DA,OAB)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(AO,OD,DC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),20)
Equal(MeasureOfAngle(DBC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAD))
Answer: 15
Theorem sequence:
triangle_property_angle_sum(1,ADO)
triangle_property_angle_sum(1,CDB)
adjacent_complementary_angle(1,CDB,BDA)
adjacent_complementary_angle(1,BDA,ADO)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, AD=CD, ∠FEB=70°, the center of circle O is O, AB is the diameter of circle O. Find the measure of ∠ABC.
问题：如图所示，AD=CD，∠FEB=70°，O是圆O的圆心，圆O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AF,FD)
Shape(DC,OCD)
Shape(DF,FO,OD)
Shape(DO,OB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BE,OEB)
Shape(BF,FE,EB)
Shape(EF,FA,OAE)
Collinear(AFOB)
Collinear(DFE)
Cocircular(O,DAEBC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(CD))
Equal(MeasureOfAngle(FEB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 40
Theorem sequence:
congruent_arc_judgment_chord_equal(1,ODA,OCD)
congruent_arc_property_measure_equal(1,ODA,OCD)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,E)
arc_property_circumference_angle_external(1,OCD,B)
arc_property_center_angle(1,ODA,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,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,DAO)
angle_addition(1,OBD,DBC)
Problem: As shown in the diagram, ∠OBA=40°, the center of circle O is O. Find the measure of ∠ACB.
问题：如图所示，∠OBA=40°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AD,DB,BA)
Shape(AB,OBA)
Shape(BD,DC,CB)
Shape(BC,OCB)
Shape(CD,DO,OA,OAC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 50
Theorem sequence:
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_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠CDA=120°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CAB.
问题：如图所示，∠CDA=120°，⊙O的圆心为O，⊙O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(DA,AC,CD)
Shape(CA,AB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_internal(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, the measure of ⌒OEC is 92, O is the center of ⊙O, CD is the diameter of ⊙O, AO is parallel to DE. Find the measure of ∠OCA.
问题：如图所示，⌒OEC的角度为92，O是圆O的圆心，CD是⊙O的直径，AO∥DE。求∠OCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OE,OEC)
Shape(OD,DE,EO)
Shape(ED,ODE)
Shape(DO,OA,OAD)
Collinear(COD)
Cocircular(O,CADE)
Conditions (CDL):
Equal(MeasureOfArc(OEC),92)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(OCA))
Answer: 23
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OEC,D)
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
Problem: As shown in the diagram, ∠ABD=32°, O is the center of circle O. Find the measure of ∠CAO.
问题：如图所示，∠ABD=32°，⊙O的圆心为O。求∠CAO的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CD,DO,OC)
Shape(CO,OD,DB,OBC)
Collinear(BDC)
Collinear(ADO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),32)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 58
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
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,CAO)
Problem: As shown in the diagram, Sub(MeasureOfAngle(DBC)=MeasureOfAngle(OAD)), the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，Sub(MeasureOfAngle(DBC)=MeasureOfAngle(OAD))，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BD,DA,OAB)
Shape(OA,AD,DO)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(Sub(MeasureOfAngle(DBC),MeasureOfAngle(OAD)),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)
adjacent_complementary_angle(1,ODC,CDB)
adjacent_complementary_angle(1,ADO,ODC)
triangle_property_angle_sum(1,CDB)
triangle_property_angle_sum(1,OAD)
Problem: As shown in the diagram, the length of ⌒OAC is equal to the length of arc OEA, ∠ABC=128°, the center of ⊙O is O. Find the measure of ∠CDE.
问题：如图所示，⌒OAC的长度与弧OEA的长度相等，∠ABC=128°，圆O的圆心为O。求∠CDE的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(DE,ODE)
Shape(AE,OEA)
Shape(AB,BC,CD,DA)
Shape(AD,DE,EA)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(LengthOfArc(OAC),LengthOfArc(OEA))
Equal(MeasureOfAngle(ABC),128)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDE))
Answer: 104
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAC,OEA)
congruent_arc_property_measure_equal(1,OAC,OEA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OEA,D)
arc_property_circumference_angle_internal(1,OAC,B)
arc_property_circumference_angle_internal(1,OCA,D)
angle_addition(1,CDA,ADE)
Problem: As shown in the diagram, ∠BOA=66°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=66°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(BC,CA,AO,OB)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),66)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 33
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BCD=140°, the center of circle O is O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=140°，O是圆O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OD,DA)
Shape(BC,CD,DO,OB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠COB=100°, the center of ⊙O is 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(CO,OB,OBC)
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, ∠ADC=75°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=75°，O是⊙O的圆心，圆O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AE,EC,CA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BE,ED,ODB)
Shape(EA,AD,DE)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),75)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 15
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠DCA=40°, O is the center of circle O, AB is the diameter of circle O, the tangent to circle O is CD. Find the measure of ∠OBD.
问题：如图所示，∠DCA=40°，O是圆O的圆心，AB是圆O的直径，⊙O的切线为CD。求∠OBD的大小。
Construction (CDL):
Shape(DC,CA,ODA)
Shape(DB,OBD)
Shape(DO,OB,BD)
Shape(AO,OD,ODA)
Shape(BA,OAB)
Collinear(CAOB)
Cocircular(O,DAB)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Goal: Value(MeasureOfAngle(OBD))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CD,O,O)
triangle_property_angle_sum(1,DOB)
triangle_property_angle_sum(1,DCO)
adjacent_complementary_angle(1,COD,DOB)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBD)
isosceles_triangle_property_angle_equal(1,OBD)
Problem: As shown in the diagram, DA=DC, ∠CBE=45°, O is the center of ⊙O. Find the measure of ∠DAC.
问题：如图所示，DA=DC，∠CBE=45°，⊙O的圆心为O。求∠DAC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AC,CD)
Shape(DC,OCD)
Shape(CA,AB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(ABE)
Cocircular(O,DABC)
Conditions (CDL):
Equal(LengthOfLine(DA),LengthOfLine(DC))
Equal(MeasureOfAngle(CBE),45)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 135/2
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBE)
arc_property_circumference_angle_internal(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
triangle_property_angle_sum(1,DAC)
Problem: As shown in the diagram, ∠BCA=36°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=36°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(BD,DA,OAB)
Shape(CD,DB,BC)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),36)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 72
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, AB=AC, ∠ABC=70°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，AB=AC，∠ABC=70°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AC,OCA)
Shape(AB,BC,CA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠CBA=100°, the center of circle O is O. Find the measure of ∠EDA.
问题：如图所示，∠CBA=100°，O是圆O的圆心。求∠EDA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(DA,OAD)
Shape(BC,OCB)
Shape(AD,DC,CB,BA)
Shape(CD,ODC)
Collinear(EDC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EDA))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
adjacent_complementary_angle(1,EDA,ADC)