SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ABO=55°, the center of circle O is O, BD⊥AD. Find the measure of ∠COB.
问题：如图所示，∠ABO=55°，圆O的圆心为O，BD垂直于AD。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BD,DA)
Shape(OD,DC,CO)
Shape(CD,DB,OBC)
Shape(AD,DO,OC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),55)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BD,AD)
Goal: Value(MeasureOfAngle(COB))
Answer: 70
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, ∠OAB=50°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=50°，圆O的圆心为O。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(OAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
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,OAB)
Problem: As shown in the diagram, the length of ⌒OAB is equal to the length of arc OCA, ∠BCA=75°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，弧OAB的长度与⌒OCA的长度相等，∠BCA=75°，O是圆O的圆心。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(LengthOfArc(OAB),LengthOfArc(OCA))
Equal(MeasureOfAngle(BCA),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OCA)
congruent_arc_property_chord_equal(1,OAB,OCA)
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, AB=10, AC=6, O is the center of circle O, the diameter of circle O is AB. Find the length of line BC.
问题：如图所示，AB=10，AC=6，O是圆O的圆心，AB是圆O的直径。求直线BC的长度。
Construction (CDL):
Shape(BC,OCB)
Shape(CB,BO,OA,AC)
Shape(CA,OAC)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),6)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, ∠BOA=38°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=38°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(DB,BC,CD)
Shape(BD,DA,OAB)
Shape(CB,OBC)
Shape(AO,OD,DC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),38)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 19
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ADC=35°, O is the center of circle O, the diameter of circle O is BC. Find the measure of ∠BCA.
问题：如图所示，∠ADC=35°，圆O的圆心为O，圆O的直径为BC。求∠BCA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(CE,ED,DC)
Shape(CD,ODC)
Shape(DE,EO,OB,OBD)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
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)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠CDA=65°, the center of circle O is O, the diameter of ⊙O is AB, CE⊥AE. Find the measure of ∠OCD.
问题：如图所示，∠CDA=65°，O是圆O的圆心，圆O的直径为AB，CE⊥AE。求∠OCD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OE,ED)
Shape(OC,CE,EO)
Shape(BE,EC,CB)
Shape(DE,EB,OBD)
Shape(BC,COB)
Shape(CO,OA,OAC)
Collinear(DEC)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),65)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CE,AE)
Goal: Value(MeasureOfAngle(OCD))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
adjacent_complementary_angle(1,BOC,COA)
triangle_property_angle_sum(1,OCE)
Problem: As shown in the diagram, ∠CDA=62°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠ACO.
问题：如图所示，∠CDA=62°，O是⊙O的圆心，圆O的直径为AB。求∠ACO的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CB,BE,EC)
Shape(CE,EO,OC)
Shape(CO,OA,AC)
Shape(CA,OAC)
Shape(AO,OE,ED,DA)
Shape(DE,EB,OBD)
Shape(AD,ODA)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),62)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ACO))
Answer: 28
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OAC,D)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,CBO)
Problem: As shown in the diagram, ∠ABC=30°, the center of circle O is O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=30°，O是圆O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, AC=2, AO=3/2, O is the center of ⊙O, AD is the diameter of circle O. Find the length of line DC.
问题：如图所示，AC=2，AO=3/2，⊙O的圆心为O，⊙O的直径为AD。求直线DC的长度。
Construction (CDL):
Shape(AB,BE,EO,OA)
Shape(AO,OE,EC,CA)
Shape(DE,EB,OBD)
Shape(CE,ED,DC)
Shape(CD,ODC)
Shape(BA,OAB)
Shape(AC,OCA)
Collinear(AOED)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(AO),3/2)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(DC))
Answer: sqrt(5)
Theorem sequence:
line_addition(1,AO,OD)
diameter_of_circle_property_right_angle(1,DCA,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
right_triangle_judgment_angle(1,DCA)
right_triangle_property_pythagorean(1,DCA)
Problem: As shown in the diagram, ∠ACD=42°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DAB.
问题：如图所示，∠ACD=42°，O是圆O的圆心，⊙O的直径为AB。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(AC,CE,EA)
Shape(DE,EO,OB,BD)
Shape(DB,OBD)
Shape(CA,OAC)
Shape(BO,OE,EC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),42)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 48
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠CAB=20°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CAB=20°，圆O的圆心为O，圆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(CAB),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(COB))
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, ∠AOB=100°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=100°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(BD,DC,CB)
Shape(BC,OCB)
Shape(AD,DB,OBA)
Shape(CD,DO,OA,OAC)
Collinear(BDO)
Collinear(ADC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠CDB=30°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠CDB=30°，O是圆O的圆心，AB是⊙O的直径。求∠AOC的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BE,ED)
Shape(EC,CO,OE)
Shape(DE,EO,OA,OAD)
Shape(CE,EB,OBC)
Shape(AO,OC,OCA)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
adjacent_complementary_angle(1,AOC,COB)
Problem: As shown in the diagram, ∠CPB=60°, the center of circle P is P, AB is the diameter of ⊙P. Find the measure of ∠CDB.
问题：如图所示，∠CPB=60°，P是圆P的圆心，圆P的直径为AB。求∠CDB的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(DE,EP,PD)
Shape(DP,PA,AD)
Shape(AP,PC,CA)
Shape(PE,EC,CP)
Shape(CE,EB,BC)
Shape(BD,PDB)
Shape(CB,PBC)
Shape(AC,PCA)
Shape(DA,PAD)
Collinear(BEPA)
Collinear(DEC)
Cocircular(P,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CPB),60)
IsCentreOfCircle(P,P)
IsDiameterOfCircle(AB,P)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,PBC,P)
arc_property_circumference_angle_external(1,PBC,D)
Problem: As shown in the diagram, ∠BOD=140°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠DCA.
问题：如图所示，∠BOD=140°，⊙O的圆心为O，AB是⊙O的直径。求∠DCA的大小。
Construction (CDL):
Shape(CA,AE,EC)
Shape(AC,OCA)
Shape(CE,EO,OB,BC)
Shape(CB,OBC)
Shape(OE,ED,DO)
Shape(DE,EA,OAD)
Shape(BO,OD,ODB)
Collinear(AOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),140)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,BOD,DOA)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠AOC=110°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠CDB.
问题：如图所示，∠AOC=110°，圆O的圆心为O，⊙O的直径为AB。求∠CDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(BE,ED,DB)
Shape(DE,EO,OA,OAD)
Shape(CO,OE,EC)
Shape(CE,EB,OBC)
Shape(AO,OC,OCA)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ⌒OBA is congruent to ⌒OAC, ∠AOB=50°, the center of ⊙O is O. Find the measure of ∠CDE.
问题：如图所示，⌒OBA全等于⌒OAC，∠AOB=50°，O是圆O的圆心。求∠CDE的大小。
Construction (CDL):
Shape(CF,FA,OAC)
Shape(AF,FG,GA)
Shape(AG,GB,OBA)
Shape(FC,CE,EF)
Shape(GF,FE,EO,OG)
Shape(BG,GO,OB)
Shape(DC,OCD)
Shape(EC,CD,DE)
Shape(BO,OE,ED,ODB)
Collinear(AGO)
Collinear(AFED)
Collinear(CFGB)
Collinear(CEO)
Cocircular(O,DBAC)
Conditions (CDL):
CongruentBetweenArc(OBA,OAC)
Equal(MeasureOfAngle(AOB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDE))
Answer: 25
Theorem sequence:
congruent_arc_property_measure_equal(1,OBA,OAC)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ⌒OBC is congruent to ⌒ODB, the diameter of ⊙O is 10, ∠CAO=32°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BOD.
问题：如图所示，⌒OBC全等于⌒ODB，⊙O的直径为10，∠CAO=32°，圆O的圆心为O，AB是⊙O的直径。求∠BOD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,OBC)
Shape(BO,OD,ODB)
Shape(DO,OA,OAD)
Collinear(AOB)
Cocircular(O,DBCA)
Conditions (CDL):
CongruentBetweenArc(OBC,ODB)
Equal(DiameterOfCircle(O),10)
Equal(MeasureOfAngle(CAO),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 64
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)
triangle_property_angle_sum(1,OCA)
flat_angle(1,AOB)
angle_addition(1,AOC,COB)
arc_property_center_angle(1,ODB,O)
arc_property_center_angle(1,OBC,O)
congruent_arc_property_measure_equal(1,OBC,ODB)
Problem: As shown in the diagram, BC=12, ∠EAC=30°, O is the center of circle O, the diameter of circle O is BD. Find the length of line BD.
问题：如图所示，BC=12，∠EAC=30°，O是圆O的圆心，BD是圆O的直径。求直线BD的长度。
Construction (CDL):
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AC,CF,FA)
Shape(FC,CE,EF)
Shape(EC,CB,BE)
Shape(EB,BO,OE)
Shape(DF,FE,EO,OD)
Shape(AF,FD,ODAs)
Shape(DO,OB,OBD)
Collinear(CEO)
Collinear(CFD)
Collinear(BEFA)
Collinear(BOD)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(LengthOfLine(BC),12)
Equal(MeasureOfAngle(EAC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(LengthOfLine(BD))
Answer: 24
Theorem sequence:
diameter_of_circle_property_right_angle(1,DCB,O)
arc_property_circumference_angle_external(1,OCB,D)
arc_property_circumference_angle_external(1,OCB,A)
sine_theorem(1,BDC)
Problem: As shown in the diagram, ∠CAB=48°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=48°，圆O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(CO,OB,OBC)
Shape(BA,OAB)
Shape(AC,OCA)
Shape(AB,BO,OC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),48)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 96
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CAO=70°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠BCO.
问题：如图所示，∠CAO=70°，O是⊙O的圆心，圆O的直径为AB。求∠BCO的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,BC)
Shape(CB,OBC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 20
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
diameter_of_circle_property_right_angle(1,BCA,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠BCE=20°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DBE.
问题：如图所示，∠BCE=20°，O是⊙O的圆心，圆O的直径为AB。求∠DBE的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BD,ODB)
Shape(BE,ED,DB)
Shape(DE,EO,OA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBE))
Answer: 70
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, ∠OBC=30°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠CAO.
问题：如图所示，∠OBC=30°，O是圆O的圆心，AB是圆O的直径。求∠CAO的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠OCA=25°, the center of circle O is O, CD is the diameter of circle O, OE⊥BE. Find the measure of ∠BOE.
问题：如图所示，∠OCA=25°，⊙O的圆心为O，CD是⊙O的直径，OE⊥BE。求∠BOE的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(OA,AE,EO)
Shape(OE,EB,BO)
Shape(CO,OB,OBC)
Shape(DE,EA,OAD)
Shape(BE,ED,ODB)
Collinear(COED)
Collinear(AEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(BOE))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
flat_angle(1,AEB)
angle_addition(1,AEO,OEB)
triangle_property_angle_sum(1,OAE)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OEB)
Problem: As shown in the diagram, ∠CAB=68°, O is the center of circle O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=68°，O是圆O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AC,OCA)
Shape(OB,BC,CO)
Shape(AB,BO,OC,CA)
Shape(CB,OBC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),68)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 22
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,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,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠OBC=40°, O is the center of ⊙O. Find the measure of ∠BAC.
问题：如图所示，∠OBC=40°，圆O的圆心为O。求∠BAC的大小。
Construction (CDL):
Shape(OB,BC,CO)
Shape(CB,BA,AC)
Shape(CA,OAC)
Shape(AB,OBA)
Shape(BO,OC,OCB)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 130
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,OBC)
round_angle(1,COB,BOC)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠ABO=30°, the center of ⊙O is O, BC is the diameter of circle O. Find the measure of ∠OCA.
问题：如图所示，∠ABO=30°，圆O的圆心为O，BC是⊙O的直径。求∠OCA的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(BA,OAB)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Collinear(BOC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠CAE=25°, the center of ⊙O is O. Find the measure of ∠EOB.
问题：如图所示，∠CAE=25°，⊙O的圆心为O。求∠EOB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,OBC)
Shape(BE,EO,OB)
Shape(EA,AO,OE)
Shape(BO,OA,OAB)
Collinear(CEO)
Collinear(AEB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAE),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EOB))
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, ∠CBA=55°, O is the center of ⊙O, AO∥DC. Find the measure of ∠DOA.
问题：如图所示，∠CBA=55°，O是⊙O的圆心，AO平行于DC。求∠DOA的大小。
Construction (CDL):
Shape(BA,AO,OC,CB)
Shape(OD,DC,CO)
Shape(DO,OA,OAD)
Shape(AB,OBA)
Shape(BC,OCB)
Shape(CD,ODC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),55)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(DOA))
Answer: 70
Theorem sequence:
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,ODC)
isosceles_triangle_property_angle_equal(1,ODC)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
parallel_property_alternate_interior_angle(2,AO,DC)
parallel_property_ipsilateral_internal_angle(1,CD,OA)
Problem: As shown in the diagram, ∠CAO=28°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠ADE.
问题：如图所示，∠CAO=28°，圆O的圆心为O，圆O的直径为AB。求∠ADE的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DA,OAD)
Shape(EO,OA,AD,DE)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),28)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADE))
Answer: 62
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, ∠ABC=130°, O is the center of circle O. Find the measure of ∠ACO.
问题：如图所示，∠ABC=130°，O是圆O的圆心。求∠ACO的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(CA,AB,BC)
Shape(OA,AC,CO)
Shape(AO,OC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACO))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
round_angle(1,AOC,COA)
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,OAC)
Problem: As shown in the diagram, ∠CBA=52°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，∠CBA=52°，O是⊙O的圆心。求∠COA的大小。
Construction (CDL):
Shape(BA,AO,OC,CB)
Shape(AB,OBA)
Shape(BC,OCB)
Shape(CO,OA,AC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),52)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 104
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠ADE=55°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠CAO.
问题：如图所示，∠ADE=55°，O是⊙O的圆心，AB是⊙O的直径。求∠CAO的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(EO,OA,AD,DE)
Shape(BE,ED,ODB)
Shape(DA,OAD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 35
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, ∠CBO=19°, ∠OAC=19°, the center of ⊙O is O. Find the measure of ∠AOB.
问题：如图所示，∠CBO=19°，∠OAC=19°，圆O的圆心为O。求∠AOB的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Shape(AC,CO,OA)
Shape(OC,CB,BO)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),19)
Equal(MeasureOfAngle(OAC),19)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 76
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,ACO)
triangle_property_angle_sum(1,OCB)
angle_addition(1,BOC,COA)
round_angle(1,AOB,BOA)
Problem: As shown in the diagram, ∠COE=70°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠EDB.
问题：如图所示，∠COE=70°，圆O的圆心为O，⊙O的直径为AB。求∠EDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BE,ED)
Shape(DE,EO,OA,OAD)
Shape(AO,OC,OCA)
Shape(OE,EC,CO)
Shape(CE,EB,OBC)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(COE),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)