SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=AC, ∠BAE=45°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠CBE.
问题：如图所示，AB=AC，∠BAE=45°，圆O的圆心为O，圆O的直径为AB。求∠CBE的大小。
Construction (CDL):
Shape(EC,CD,OED)
Shape(DB,BE,OED)
Shape(EA,OAE)
Shape(BD,ODB)
Shape(AE,EB,BO,OA)
Shape(AO,OB,OBA)
Collinear(AEC)
Collinear(CDB)
Collinear(AOB)
Cocircular(O,AEDB)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(BAE),45)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBE))
Answer: 45/2
Theorem sequence:
diameter_of_circle_property_right_angle(1,AEB,O)
triangle_property_angle_sum(1,AEB)
triangle_property_angle_sum(1,CBA)
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_property_angle_equal(1,ACB)
angle_addition(1,CBE,EBA)
Problem: As shown in the diagram, ∠CAD=50°, ∠DBO=30°, O is the center of circle O. Find the measure of ∠CDB.
问题：如图所示，∠CAD=50°，∠DBO=30°，O是⊙O的圆心。求∠CDB的大小。
Construction (CDL):
Shape(BD,DA,OAB)
Shape(DB,BO,OD)
Shape(AD,DO,OC,CA)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Collinear(ADB)
Collinear(DOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),50)
Equal(MeasureOfAngle(DBO),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
adjacent_complementary_angle(1,COB,BOD)
triangle_property_angle_sum(1,DBO)
Problem: As shown in the diagram, ∠BCO=45°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=45°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(BO,OC,CA,AB)
Shape(BC,CO,OB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),45)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 45
Theorem sequence:
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,BCO)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠OCA=30°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠COB.
问题：如图所示，∠OCA=30°，O是圆O的圆心，⊙O的直径为AB。求∠COB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 60
Theorem sequence:
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CAB=36°, O is the center of circle O, AB is the diameter of ⊙O, AE⊥CE. Find the measure of ∠BCD.
问题：如图所示，∠CAB=36°，圆O的圆心为O，AB是圆O的直径，AE⊥CE。求∠BCD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OE,EC,CA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(AD,DE,EO,OA)
Shape(BE,ED,ODB)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),36)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(BCD))
Answer: 36
Theorem sequence:
triangle_property_angle_sum(1,AEC)
diameter_of_circle_property_right_angle(1,BCA,O)
angle_addition(1,BCE,ECA)
Problem: As shown in the diagram, ∠CFB=20°, the center of circle A is A, DG is the perpendicular bisector of AB. Find the measure of ∠DBC.
问题：如图所示，∠CFB=20°，⊙A的圆心为A，DG垂直平分AB。求∠DBC的大小。
Construction (CDL):
Shape(FG,GA,AD,DE,AEF)
Shape(BF,AFB)
Shape(FB,BH,HF)
Shape(FH,HG,GF)
Shape(AG,GD,DA)
Shape(GH,HI,ID,DG)
Shape(HB,BI,IH)
Shape(IB,BC,CI)
Shape(CB,ABC)
Shape(DI,IC,CD)
Shape(ED,DC,ACE)
Collinear(FGDE)
Collinear(FHIC)
Collinear(AGHB)
Collinear(ADC)
Collinear(BID)
Cocircular(A,FBCE)
Conditions (CDL):
Equal(MeasureOfAngle(CFB),20)
IsCentreOfCircle(A,A)
IsPerpendicularBisectorOfLine(DG,AB)
Goal: Value(MeasureOfAngle(DBC))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,ABC,F)
arc_property_center_angle(1,ABC,A)
perpendicular_bisector_property_distance_equal(1,DG,AB)
radius_of_circle_property_length_equal(1,AB,A)
radius_of_circle_property_length_equal(1,AC,A)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
angle_addition(1,ABD,DBC)
Problem: As shown in the diagram, ∠DAB=30°, O is the center of ⊙O, the diameter of circle O is AB, OC is parallel to BD. Find the measure of ∠CBD.
问题：如图所示，∠DAB=30°，O是圆O的圆心，AB是⊙O的直径，OC平行于BD。求∠CBD的大小。
Construction (CDL):
Shape(AE,EC,OCA)
Shape(EA,AO,OE)
Shape(CE,EF,FC)
Shape(CF,FD,ODC)
Shape(FE,EO,OB,BF)
Shape(FB,BD,DF)
Shape(DB,OBD)
Collinear(AEFD)
Collinear(CEO)
Collinear(CFB)
Collinear(AOB)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(OC,BD)
Goal: Value(MeasureOfAngle(CBD))
Answer: 30
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,ABD)
parallel_property_ipsilateral_internal_angle(1,OC,BD)
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,COB)
angle_addition(1,OBC,CBD)
Problem: As shown in the diagram, BE=DE, ∠DBE=40°, the center of circle O is O. Find the measure of ∠CAE.
问题：如图所示，BE=DE，∠DBE=40°，O是圆O的圆心。求∠CAE的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,OBC)
Shape(BE,ED,DB)
Shape(DE,EA,OAD)
Shape(BD,ODB)
Collinear(CED)
Collinear(AEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(BE),LengthOfLine(DE))
Equal(MeasureOfAngle(DBE),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAE))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,EDB)
isosceles_triangle_property_angle_equal(1,EDB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠ABC=24°, O is the center of circle O, the diameter of ⊙O is AB, AC∥DO. Find the measure of ∠BCD.
问题：如图所示，∠ABC=24°，O是⊙O的圆心，AB是⊙O的直径，AC平行于DO。求∠BCD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AE,EC,CA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EA,AD)
Shape(DA,OAD)
Shape(DO,OE,ED)
Shape(BO,OD,ODB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),24)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AC,DO)
Goal: Value(MeasureOfAngle(BCD))
Answer: 57
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
parallel_property_alternate_interior_angle(1,AC,DO)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OAD)
isosceles_triangle_property_angle_equal(1,OAD)
triangle_property_angle_sum(1,OAD)
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠FDE=50°, O is the center of circle O, AE is perpendicular to DE, OF is perpendicular to DF. Find the measure of ∠ACB.
问题：如图所示，∠FDE=50°，O是⊙O的圆心，AE垂直于DE，OF⊥DF。求∠ACB的大小。
Construction (CDL):
Shape(AE,ED,ODA)
Shape(DE,EO,OF,FD)
Shape(DF,FB,OBD)
Shape(AC,CB,BO,OA)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(AEO)
Collinear(OFB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(FDE),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,DE)
PerpendicularBetweenLine(OF,DF)
Goal: Value(MeasureOfAngle(ACB))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,AED,DEO)
quadrilateral_property_angle_sum(1,DEOF)
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠ACO=50°, the center of circle O is O, AB is the diameter of circle O. Find the measure of ∠CBO.
问题：如图所示，∠ACO=50°，O是圆O的圆心，AB是⊙O的直径。求∠CBO的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CO,OA)
Shape(OC,CB,BO)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACO),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBO))
Answer: 40
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,ACO)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
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是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BO,OD,DA,OAB)
Shape(OB,BD,DO)
Shape(DC,CA,AD)
Shape(AC,OCA)
Shape(CD,DB,OBC)
Collinear(ODC)
Collinear(ADB)
Cocircular(O,BCA)
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, ∠AMC=90°, ∠PNA=60°, O is the center of circle O, the diameter of ⊙O is CD, M is the midpoint of segment AB. Find the measure of ∠PDC.
问题：如图所示，∠AMC=90°，∠PNA=60°，O是圆O的圆心，圆O的直径为CD，M平分线段AB。求∠PDC的大小。
Construction (CDL):
Shape(AN,NC,OCA)
Shape(NM,MC,CN)
Shape(CM,MB,OBC)
Shape(PN,NA,OAP)
Shape(NP,PD,DO,OM,MN)
Shape(BM,MO,OD,ODB)
Shape(DP,OPD)
Collinear(ANMB)
Collinear(CNP)
Collinear(CMOD)
Cocircular(O,APDBC)
Conditions (CDL):
Equal(MeasureOfAngle(AMC),90)
Equal(MeasureOfAngle(PNA),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
IsMidpointOfLine(M,AB)
Goal: Value(MeasureOfAngle(PDC))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,CPD,O)
vertical_angle(1,CNM,PNA)
triangle_property_angle_sum(1,NMC)
triangle_property_angle_sum(1,CPD)
Problem: As shown in the diagram, ∠EAD=35°, the center of ⊙O is O, AB is the diameter of circle O, DE⊥AE. Find the measure of ∠AOC.
问题：如图所示，∠EAD=35°，⊙O的圆心为O，圆O的直径为AB，DE⊥AE。求∠AOC的大小。
Construction (CDL):
Shape(BE,ED,ODB)
Shape(DE,EO,OA,AD)
Shape(DA,OAD)
Shape(CE,EB,OBC)
Shape(EC,CO,OE)
Shape(AO,OC,OCA)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(DE,AE)
Goal: Value(MeasureOfAngle(AOC))
Answer: 110
Theorem sequence:
triangle_property_angle_sum(1,DEA)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, the length of ⌒OAB is equal to the length of arc OBC, ∠BOA=72°, O is the center of ⊙O. Find the measure of ∠OAC.
问题：如图所示，⌒OAB的长度与弧OBC的长度相等，∠BOA=72°，O是⊙O的圆心。求∠OAC的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(AD,DO,OA)
Shape(DA,AB,BD)
Shape(DB,BC,CD)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OBC))
Equal(MeasureOfAngle(BOA),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 18
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_center_angle(1,OAB,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,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
angle_addition(1,OAC,CAB)
Problem: As shown in the diagram, ∠ADC=36°, O is the center of ⊙O, the diameter of ⊙O is BC. Find the measure of ∠BCA.
问题：如图所示，∠ADC=36°，⊙O的圆心为O，⊙O的直径为BC。求∠BCA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(BO,OE,EA,AB)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(DE,EO,OB,OBD)
Shape(ED,DC,CE)
Shape(CD,ODC)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),36)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 54
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AB=AC, ∠DBC=20°, O is the center of circle O. Find the measure of ∠DAE.
问题：如图所示，AB=AC，∠DBC=20°，⊙O的圆心为O。求∠DAE的大小。
Construction (CDL):
Shape(DA,AE,ODE)
Shape(ODE,EB,BD)
Shape(BE,OEB)
Shape(BO,OC,CD,DB)
Shape(DC,OCD)
Collinear(AEB)
Collinear(ADC)
Collinear(BOC)
Cocircular(O,CDEB)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(DBC),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAE))
Answer: 40
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,COB,O)
diameter_of_circle_property_right_angle(1,CDB,O)
triangle_property_angle_sum(1,BCD)
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
Problem: As shown in the diagram, ∠BAC=40°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠CDA.
问题：如图所示，∠BAC=40°，O是圆O的圆心，AB是圆O的直径。求∠CDA的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(CE,EA,AC)
Shape(CB,BO,OE,EC)
Shape(AE,ED,DA)
Shape(AD,ODA)
Shape(BC,OCB)
Shape(DE,EO,OB,OBD)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠COB=68°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠CDB.
问题：如图所示，∠COB=68°，O是圆O的圆心，AB是圆O的直径。求∠CDB的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(DE,EC,CD)
Shape(AO,OE,ED,ODA)
Shape(EO,OB,BE)
Shape(CE,EB,OBC)
Collinear(DEB)
Collinear(OEC)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),68)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 34
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠ABD=55°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=55°，O是⊙O的圆心，AB是⊙O的直径。求∠DCB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AO,OE,ED,DA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(CE,EO,OA,OAC)
Shape(EC,CB,BE)
Shape(BC,OCB)
Collinear(AOEB)
Collinear(CED)
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, ∠COA=110°, the center of circle O is O. Find the measure of ∠ADC.
问题：如图所示，∠COA=110°，O是圆O的圆心。求∠ADC的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(AD,DC,CO,OA)
Shape(DA,OAD)
Shape(CD,ODC)
Cocircular(O,ADC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 125
Theorem sequence:
round_angle(1,AOC,COA)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BCD=70°, O is the center of ⊙O, BO is perpendicular to AO, OE is perpendicular to AE. Find the measure of ∠EAO.
问题：如图所示，∠BCD=70°，⊙O的圆心为O，BO垂直于AO，OE垂直于AE。求∠EAO的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OE,EA)
Shape(AE,ED,ODA)
Shape(BC,CD,DE,EO,OB)
Shape(CB,OBC)
Shape(DC,OCD)
Collinear(OED)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),70)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BO,AO)
PerpendicularBetweenLine(OE,AE)
Goal: Value(MeasureOfAngle(EAO))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_center_angle(1,ODB,O)
angle_addition(1,BOA,AOE)
triangle_property_angle_sum(1,AOE)
Problem: As shown in the diagram, the length of ⌒OCD is equal to the length of ⌒ODB, ∠ABC=70°, ∠BCA=30°, the center of circle O is O. Find the measure of ∠DBC.
问题：如图所示，弧OCD的长度与⌒ODB的长度相等，∠ABC=70°，∠BCA=30°，圆O的圆心为O。求∠DBC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(AE,ED,ODA)
Shape(EB,BC,CE)
Shape(DE,EC,CD)
Shape(DC,OCD)
Shape(CB,OBC)
Collinear(AEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OCD),LengthOfArc(ODB))
Equal(MeasureOfAngle(ABC),70)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBC))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BCD)
congruent_arc_judgment_length_equal(1,ODB,OCD)
congruent_arc_property_measure_equal(1,ODB,OCD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
arc_property_circumference_angle_external(1,OCD,B)
arc_property_circumference_angle_external(1,ODB,C)
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是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(CO,OB,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, ∠BAD=36°, ∠DBA=64°, the center of ⊙O is O. Find the measure of ∠OCD.
问题：如图所示，∠BAD=36°，∠DBA=64°，O是⊙O的圆心。求∠OCD的大小。
Construction (CDL):
Shape(CO,OD,DA,OAC)
Shape(OC,CD,DO)
Shape(AD,DB,BA)
Shape(AB,OBA)
Shape(BD,DC,OCB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),36)
Equal(MeasureOfAngle(DBA),64)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCD))
Answer: 28
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
triangle_property_angle_sum(1,OCD)
triangle_property_angle_sum(1,ADB)
vertical_angle(1,ADB,CDO)
Problem: As shown in the diagram, Add(MeasureOfAngle(DCB)=MeasureOfAngle(AOD)), the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠DCB.
问题：如图所示，Add(MeasureOfAngle(DCB)=MeasureOfAngle(AOD))，⊙O的圆心为O，AB是⊙O的直径。求∠DCB的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(CD,ODC)
Shape(CE,ED,DC)
Shape(DE,EB,OBD)
Shape(DO,OB,BE)
Shape(BO,OA,OAB)
Collinear(CEB)
Collinear(OED)
Collinear(AOB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DCB),MeasureOfAngle(AOD)),145)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,AOD,DOB)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠ABC=75°, O is the center of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=75°，O是圆O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CO,OA)
Shape(CB,OBC)
Shape(AO,OC,OCA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 150
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠FDB=30°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠AOE.
问题：如图所示，∠FDB=30°，O是⊙O的圆心，AB是⊙O的直径。求∠AOE的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BF,FD)
Shape(EF,FB,OBE)
Shape(FE,EO,OF)
Shape(DF,FO,OA,OAD)
Shape(AO,OE,OEA)
Collinear(BFOA)
Collinear(DFE)
Cocircular(O,BEAD)
Conditions (CDL):
Equal(MeasureOfAngle(FDB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBE,O)
arc_property_circumference_angle_external(1,OBE,D)
adjacent_complementary_angle(1,AOE,EOB)
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，⊙O的直径为BC。求∠CAO的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(BO,OA,AB)
Shape(AO,OE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(DE,EO,OB,OBD)
Shape(DC,CE,ED)
Shape(CD,ODC)
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:
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)
diameter_of_circle_property_right_angle(1,CAB,O)
angle_addition(1,CAO,OAB)
Problem: As shown in the diagram, ∠POA=40°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠BCP.
问题：如图所示，∠POA=40°，⊙O的圆心为O，⊙O的直径为AB。求∠BCP的大小。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BO,OP,OPB)
Shape(PO,OD,DP)
Shape(PD,DA,OAP)
Collinear(ADOB)
Collinear(CDP)
Cocircular(O,APBC)
Conditions (CDL):
Equal(MeasureOfAngle(POA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCP))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,BOP,POA)
arc_property_center_angle(1,OPB,O)
arc_property_circumference_angle_external(1,OPB,C)
Problem: As shown in the diagram, ∠ABO=20°, ∠ODA=15°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠ABO=20°，∠ODA=15°，⊙O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OD)
Shape(AB,BO,OA)
Shape(DO,OB,OBD)
Shape(BA,OAB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),20)
Equal(MeasureOfAngle(ODA),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,OAB)
Problem: As shown in the diagram, ∠BCD=50°, O is the center of circle O. Find the measure of ∠BAD.
问题：如图所示，∠BCD=50°，圆O的圆心为O。求∠BAD的大小。
Construction (CDL):
Shape(BE,ED,ODB)
Shape(CB,OBC)
Shape(CE,EB,BC)
Shape(EA,AD,DE)
Shape(DA,OAD)
Shape(AE,EC,OCA)
Collinear(BEA)
Collinear(CED)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_external(1,ODB,A)
Problem: As shown in the diagram, ∠OAB=30°, O is the center of circle O, CD is the diameter of circle O, AO∥CB. Find the measure of ∠AOD.
问题：如图所示，∠OAB=30°，O是⊙O的圆心，⊙O的直径为CD，AO∥CB。求∠AOD的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CB,BE,EC)
Shape(CE,EA,OAC)
Shape(AE,EO,OA)
Shape(AO,OD,ODA)
Shape(DO,OE,EB,OBD)
Collinear(CEOD)
Collinear(AEB)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AO,CB)
Goal: Value(MeasureOfAngle(AOD))
Answer: 120
Theorem sequence:
parallel_property_alternate_interior_angle(1,AO,CB)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
adjacent_complementary_angle(1,COA,AOD)
Problem: As shown in the diagram, ∠ACB=35°, the center of circle O is O. Find the measure of ∠AOB.
问题：如图所示，∠ACB=35°，O是圆O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CE,EA)
Shape(AE,EB,OBA)
Shape(EO,OB,BE)
Shape(BO,OE,EC,OCB)
Collinear(CEB)
Collinear(AEO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 70
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)