SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠OAB=26°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=26°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(OA,AB,BO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),26)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 64
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,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=70°, O is the center of circle O. Find the measure of ∠COB.
问题：如图所示，∠CAB=70°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AB,BO,OC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BCA=28°, O is the center of circle O. Find the measure of ∠ABO.
问题：如图所示，∠BCA=28°，圆O的圆心为O。求∠ABO的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(AD,DO,OA)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),28)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 62
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, ∠CAB=25°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=25°，⊙O的圆心为O，AB是⊙O的直径。求∠ADC的大小。
Construction (CDL):
Shape(CA,AO,OE,EC)
Shape(AD,DE,EO,OA)
Shape(CE,EB,BC)
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(CB,OBC)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠DBA=60°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠BCD.
问题：如图所示，∠DBA=60°，O是⊙O的圆心，⊙O的直径为AB。求∠BCD的大小。
Construction (CDL):
Shape(AD,DE,EO,OA)
Shape(ED,DB,BE)
Shape(EB,BC,CE)
Shape(AO,OE,EC,OCA)
Shape(DA,OAD)
Shape(BD,ODB)
Shape(CB,OBC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 30
Theorem sequence:
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠BEO=90°, ∠CDA=26°, the center of circle O is O. Find the measure of ∠AOB.
问题：如图所示，∠BEO=90°，∠CDA=26°，O是圆O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(OB,BE,EO)
Shape(EA,AF,FE)
Shape(DF,FC,CD)
Shape(AE,EB,OBA)
Shape(CF,FA,OAC)
Shape(DC,OCD)
Shape(BO,OE,EF,FD,ODB)
Collinear(OEA)
Collinear(BEFC)
Collinear(DFA)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(BEO),90)
Equal(MeasureOfAngle(CDA),26)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 52
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
circle_property_chord_perpendicular_bisect_arc(1,OBC,OEA)
congruent_arc_judgment_length_equal(1,OAC,OBA)
congruent_arc_property_measure_equal(1,OAC,OBA)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠OBC=18°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=18°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(BC,CO,OB)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC.OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),18)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 72
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ABD=110°. Find the measure of ∠BDE minus the measure of ∠BCD.
问题：如图所示，∠ABD=110°。求∠BDE的大小减去∠BCD的大小。
Construction (CDL):
Shape(AB,BD)
Shape(BC,CD,DB)
Shape(BD,DE)
Collinear(CDE)
Collinear(ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),110)
Goal: Value(Sub(MeasureOfAngle(BDE),MeasureOfAngle(BCD)))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,ABD,DBC)
adjacent_complementary_angle(1,CDB,BDE)
triangle_property_angle_sum(1,BCD)
Problem: As shown in the diagram, the length of arc OAB is equal to the length of ⌒OBC, ∠BOA=60°, O is the center of ⊙O, DB is the diameter of ⊙O. Find the measure of ∠CDB.
问题：如图所示，弧OAB的长度与弧OBC的长度相等，∠BOA=60°，圆O的圆心为O，DB是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(BO,OA,OAB)
Shape(CD,DO,OB,OBC)
Shape(DC,OCD)
Collinear(DOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OBC))
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BOA=110°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=110°，⊙O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(DA,AO,OB,BD)
Shape(OA,AB,BO)
Shape(AC,CB,BA)
Shape(AD,ODA)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(DB,OBD)
Cocircular(O,ACB,D)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=76°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=76°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(CE,EB,BC)
Shape(AO,OE,EC,OCA)
Shape(BE,EA,OAB)
Shape(CB,OBC)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),76)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 38
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OAB=45°, the center of circle O is O. Find the measure of ∠ACB.
问题：如图所示，∠OAB=45°，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(OA,AB,BO)
Shape(AC,CB,BA)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BC,OCB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),45)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 135
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,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, CA=CD, ∠DCA=30°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠CAB.
问题：如图所示，CA=CD，∠DCA=30°，⊙O的圆心为O，⊙O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(CA,AO,OE,EC)
Shape(AD,DE,EO,OA)
Shape(CE,EB,BC)
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(CB,OBC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(CA),LengthOfLine(CD))
Equal(MeasureOfAngle(DCA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 15
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CAD)
isosceles_triangle_property_angle_equal(1,CAD)
triangle_property_angle_sum(1,CAD)
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,BCA)
Problem: As shown in the diagram, ∠CAB=70°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠BCO.
问题：如图所示，∠CAB=70°，O是圆O的圆心，⊙O的直径为AB。求∠BCO的大小。
Construction (CDL):
Shape(CA,AO,OC)
Shape(CO,OB,BC)
Shape(AC,OCA)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 20
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
Problem: As shown in the diagram, ∠ABC=50°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠CAB.
问题：如图所示，∠ABC=50°，圆O的圆心为O，⊙O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(AO,OB,BC,CA)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, AD=DB, CD=8, OA=5, OA=RadiusOfCircle(O), ∠ADO=90°, the center of ⊙O is O. Find the length of line AB.
问题：如图所示，AD=DB，CD=8，OA=5，OA=RadiusOfCircle(O)，∠ADO=90°，⊙O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(OA,AD,DO)
Shape(CO,OD,DB,OBC)
Collinear(ADB)
Collinear(COD)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(DB))
Equal(LengthOfLine(CD),8)
Equal(LengthOfLine(OA),5)
Equal(LengthOfLine(OA),RadiusOfCircle(O))
Equal(MeasureOfAngle(ADO),90)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
line_addition(1,CO,OD)
right_triangle_judgment_angle(1,ADO)
right_triangle_property_pythagorean(1,ADO)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠ABD=62°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠DCB.
问题：如图所示，∠ABD=62°，⊙O的圆心为O，AB是圆O的直径。求∠DCB的大小。
Construction (CDL):
Shape(AO,OE,ED,DA)
Shape(EC,CB,BE)
Shape(EB,BD,DE)
Shape(AD,ODA)
Shape(CE,EO,OA,OAC)
Shape(BC,OCB)
Shape(DB,OBD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),62)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 28
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,BDA)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠BOC=40°, O is the center of circle O, CA∥BO. Find the measure of ∠ABO.
问题：如图所示，∠BOC=40°，圆O的圆心为O，CA平行于BO。求∠ABO的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(EB,BO,OE)
Shape(CA,OAC)
Shape(BE,EC,OCB)
Shape(AE,EO,OB,OBA)
Collinear(CEO)
Collinear(AEB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),40)
IsCentreOfCircle(O,O)
ParallelBetweenLine(CA,BO)
Goal: Value(MeasureOfAngle(ABO))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
parallel_property_alternate_interior_angle(2,CA,BO)
Problem: As shown in the diagram, ∠ABD=42°, 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，∠ABD=42°，O是圆O的圆心，⊙O的直径为AB。求∠BCD的大小。
Construction (CDL):
Shape(DB,BC,CD)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(AO,OB,BD,DA)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),42)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 132
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,BDA)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠COA=100°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠COA=100°，圆O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(DA,AO,OC,CD)
Shape(OA,AB,BC,CO)
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠BOC=50°, the center of ⊙O is O, OB∥AC. Find the measure of ∠ABO.
问题：如图所示，∠BOC=50°，圆O的圆心为O，OB∥AC。求∠ABO的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(OE,EB,BO)
Shape(AC,CE,EA)
Shape(EC,CB,BE)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(AEB)
Collinear(OEC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(OB,AC)
Goal: Value(MeasureOfAngle(ABO))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
parallel_property_alternate_interior_angle(2,CA,BO)
Problem: As shown in the diagram, ∠BCE=40°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠DBA.
问题：如图所示，∠BCE=40°，O是圆O的圆心，⊙O的直径为AB。求∠DBA的大小。
Construction (CDL):
Shape(AD,DO,OA)
Shape(OD,DE,EO)
Shape(ED,DB,BE)
Shape(EB,BC,CE)
Shape(AO,OE,EC,OCA)
Shape(DA,OAD)
Shape(BD,ODB)
Shape(CB,OBC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_external(1,ODB,A)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,DBA)
Problem: As shown in the diagram, ∠BOA=40°, O is the center of circle O, the diameter of circle O is DC, DC∥AB. Find the measure of ∠CAB.
问题：如图所示，∠BOA=40°，⊙O的圆心为O，DC是⊙O的直径，DC平行于AB。求∠CAB的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(EA,AB,BE)
Shape(AO,OD,ODA)
Shape(CE,EB,OBC)
Shape(OE,EC,CO)
Shape(DO,OC,OCD)
Collinear(DOC)
Collinear(AEC)
Collinear(OEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DC,O)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(CAB))
Answer: 35
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,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
parallel_property_collinear_extend(3,DC,AB,O)
parallel_property_alternate_interior_angle(1,OC,AB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOA=60°, O is the center of circle O. Find the measure of ∠BPA.
问题：如图所示，∠BOA=60°，圆O的圆心为O。求∠BPA的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(BP,OA,OAB)
Shape(AP,OPA)
Shape(PB,OBP)
Cocircular(O,ABP)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BPA))
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, ∠CAB=65°, O is the center of circle O. Find the measure of ∠BDC.
问题：如图所示，∠CAB=65°，⊙O的圆心为O。求∠BDC的大小。
Construction (CDL):
Shape(CA,AB,BD,DC)
Shape(CD,ODC)
Shape(AC,OCA)
Shape(BA,OAB)
Shape(DB,OBD)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),65)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 115
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_internal(1,OBC,D)
Problem: As shown in the diagram, ∠ADC=33°, O is the center of ⊙O, BC is the diameter of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠ADC=33°，O是圆O的圆心，圆O的直径为BC。求∠BCA的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(AE,EC,CA)
Shape(ED,DC,CE)
Shape(BA,OAB)
Shape(DE,EO,OB,OBD)
Shape(CD,ODC)
Shape(AC,OCA)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),33)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 57
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,CAB,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠ABD=58°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=58°，圆O的圆心为O，⊙O的直径为AB。求∠DCB的大小。
Construction (CDL):
Shape(AO,OE,ED,DA)
Shape(DE,EB,BD)
Shape(EC,CB,BE)
Shape(CE,EO,OA,OAC)
Shape(BC,OCB)
Shape(DB,OBD)
Shape(AD,ODA)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),58)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 32
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,BDA)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠BOA=50°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=50°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Shape(AO,OE,EC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(AEC)
Collinear(OEB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠AFC=36°, O is the center of circle O, AE is the median of triangle ABC, E is the midpoint of segment BC, △CAB is a right △. Find the measure of ∠ABC.
问题：如图所示，∠AFC=36°，圆O的圆心为O，AE是三角形ABC的中线，E是线段BC的中点，CA和AB是直角三角形CAB的两个直角边。求∠ABC的大小。
Construction (CDL):
Shape(GA,OAG)
Shape(EH,HA,AE)
Shape(EA,AG,OGE)
Shape(AH,HC,CA)
Shape(AC,OCA)
Shape(FH,HE,OEF)
Shape(HF,FC,CH)
Shape(CF,OFC)
Shape(GB,BE,OGE)
Collinear(BGA)
Collinear(BEHC)
Collinear(FHA)
Cocircular(O,AGEFC)
Conditions (CDL):
Equal(MeasureOfAngle(AFC),36)
IsCentreOfCircle(O,O)
IsMedianOfTriangle(AE,ABC)
IsMidpointOfLine(E,BC)
RightTriangle(CAB)
Goal: Value(MeasureOfAngle(ABC))
Answer: 18
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,F)
arc_property_circumference_angle_external(1,OCA,E)
right_triangle_property_length_of_median(1,CAB,E)
adjacent_complementary_angle(1,BEA,AEC)
line_addition(1,BE,EC)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
triangle_property_angle_sum(1,EAB)
Problem: As shown in the diagram, ∠CAB=35°, O is the center of circle O, DC is the diameter of circle O. Find the measure of ∠BCD.
问题：如图所示，∠CAB=35°，O是⊙O的圆心，圆O的直径为DC。求∠BCD的大小。
Construction (CDL):
Shape(DO,OC,OCD)
Shape(AE,ED,ODA)
Shape(AB,BE,EA)
Shape(BA,OAB)
Shape(DE,EC,CO,OD)
Shape(EB,BC,CE)
Shape(CB,OBC)
Collinear(DOC)
Collinear(DEB)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DC,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
diameter_of_circle_property_right_angle(1,DBC,O)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠CAB=55°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=55°，O是圆O的圆心，⊙O的直径为AB。求∠ADC的大小。
Construction (CDL):
Shape(CE,EO,OB,BC)
Shape(CA,AE,EC)
Shape(AD,DE,EA)
Shape(CB,OBC)
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BO,OE,ED,ODB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠DBC=40°, O is the center of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠DBC=40°，O是圆O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(DB,BC,CD)
Shape(AD,DC,OCA)
Shape(BD,DO,OA,OAB)
Shape(CB,OBC)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, CO=8, EO=6, the center of ⊙A is A, EC is the diameter of ⊙A. Find the value of cos(CB).
问题：如图所示，CO=8，EO=6，圆A的圆心为A，圆A的直径为EC。求cos(CB)的值。
Construction (CDL):
Shape(CA,AE,ED,DC)
Shape(CD,DO,OC)
Shape(DB,BO,OD)
Shape(EA,AC,ACE)
Shape(BD,DE,AEB)
Shape(OB,ABO)
Shape(CO,AOC)
Collinear(EAC)
Collinear(EDO)
Collinear(CDB)
Cocircular(A,EBOC)
Conditions (CDL):
Equal(LengthOfLine(CO),8)
Equal(LengthOfLine(EO),6)
IsCentreOfCircle(A,A)
IsDiameterOfCircle(EC,A)
Goal: Value(Cos(MeasureOfAngle(CBO)))
Answer: 3/5
Theorem sequence:
diameter_of_circle_property_right_angle(1,EOC,A)
right_triangle_judgment_angle(1,EOC)
right_triangle_property_pythagorean(1,EOC)
cosine_theorem(1,EOC)
arc_property_circumference_angle_external(1,AOC,E)
arc_property_circumference_angle_external(1,AOC,B)
Problem: As shown in the diagram, ∠ADC=36°, O is the center of circle O, BC is the diameter of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠ADC=36°，圆O的圆心为O，圆O的直径为BC。求∠BCA的大小。
Construction (CDL):
Shape(AB,BO,OE,EA)
Shape(AE,EC,CA)
Shape(ED,DC,CE)
Shape(BA,OAB)
Shape(DE,EO,OB,OBD)
Shape(CD,ODC)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
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,D)
arc_property_circumference_angle_external(1,OCA,B)
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠CAB=30°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=30°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OB,BD,DO)
Shape(DC,CA,AD)
Shape(BO,OD,DA,OAB)
Shape(AC,OCA)
Shape(CD,DB,OBC)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠AOB=100°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠AOB=100°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OB,BC,CA,AO)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_internal(1,OBA,C)
Problem: As shown in the diagram, ∠CDA=54°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠BAC.
问题：如图所示，∠CDA=54°，⊙O的圆心为O，圆O的直径为AB。求∠BAC的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(AC,CE,EA)
Shape(EC,CB,BO,OE)
Shape(DE,EO,OB,OBD)
Shape(AD,ODA)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),54)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 36
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)