SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠DOA=20°, the center of ⊙O is O, AO is parallel to DC. Find the measure of ∠CBA.
问题：如图所示，∠DOA=20°，⊙O的圆心为O，AO平行于DC。求∠CBA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OC,CB)
Shape(OD,DC,CO)
Shape(DO,OA,OAD)
Shape(CD,ODC)
Shape(BC,OCB)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),20)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(CBA))
Answer: 80
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DC)
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)
triangle_property_angle_sum(1,ODC)
angle_addition(1,COD,DOA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of ⊙O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=50°，O是⊙O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(CO,OB,BC)
Shape(BO,OC,CA,AB)
Shape(AC,OCA)
Shape(BA,OAB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,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,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠COB=100°, the center of circle O is O. Find the measure of ∠OAB.
问题：如图所示，∠COB=100°，O是⊙O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(CO,OB,OBC)
Shape(OA,AB,BO)
Shape(BA,OAB)
Shape(AC,OCA)
Collinear(AOC)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 50
Theorem sequence:
flat_angle(1,COA)
angle_addition(1,COB,BOA)
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, ∠AOB=62°, the center of ⊙O is O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=62°，⊙O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BO,OA,AC,CB)
Shape(CA,OAC)
Shape(BC,OCB)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),62)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 31
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
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⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Shape(BA,OAB)
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, the measure of ⌒OBA is 300, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，弧OBA的角度为300，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Shape(AC,OCA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfArc(OBA),300)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 30
Theorem sequence:
round_arc(1,OAB,OBA)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ABC=130°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，∠ABC=130°，O是⊙O的圆心。求∠COA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OA,AB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OAC,O)
round_arc(1,OAC,OCA)
Problem: As shown in the diagram, ∠BOA=50°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=50°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OD,DB,BO)
Shape(DC,CA,AD)
Shape(BD,DA,AB)
Shape(BA,OAB)
Shape(AC,OCA)
Shape(OC,CD,DO)
Shape(CO,OB,OBC)
Collinear(ODA)
Collinear(CDB)
Cocircular(O,BCA)
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, ∠CAB=40°, the center of ⊙O is O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=40°，圆O的圆心为O。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(EA,AD,DE)
Shape(BE,ED,ODB)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 50
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, DA=DC, ∠CAB=30°, the center of circle O is O. Find the measure of ∠DAC.
问题：如图所示，DA=DC，∠CAB=30°，圆O的圆心为O。求∠DAC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(DA,AC,CD)
Shape(CA,AB,BC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(LengthOfLine(DA),LengthOfLine(DC))
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 30
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
round_arc(1,OAC,OCA)
arc_property_circumference_angle_external(1,OCA,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=112°, O is the center of ⊙O. Find the measure of ∠AOB.
问题：如图所示，∠BCA=112°，O是圆O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CB,OBC)
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),112)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 136
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
round_arc(1,OBA,OAB)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠CBA=30°, CD bisects ∠ACB, the center of ⊙O is O. Find the measure of ∠DAC.
问题：如图所示，∠CBA=30°，CD是∠ACB的角平分线，O是圆O的圆心。求∠DAC的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(AD,ODA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(EA,AC,CE)
Shape(CA,OAC)
Shape(BE,EC,CB)
Shape(BC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),30)
IsBisectorOfAngle(CD,ACB)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 105
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
angle_addition(1,ACE,ECB)
triangle_property_angle_sum(1,DAC)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠ADC=20°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=20°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(EA,AD,DE)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 70
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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, ∠CAD=25°, the center of circle C is C, ∠BCA is the right angle of the right triangle BCA. Find the measure of ∠DCE.
问题：如图所示，∠CAD=25°，圆C的圆心为C，BC和CA是直角三角形BCA的两个直角边。求∠DCE的大小。
Construction (CDL):
Shape(BC,CD,DB)
Shape(DC,CA,AD)
Shape(BD,CDB)
Shape(EC,CB,CBE)
Collinear(CEA)
Collinear(BDA)
Cocircular(C,EDB)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),25)
IsCentreOfCircle(C,C)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(DCE))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,BCD)
triangle_property_angle_sum(1,BCA)
radius_of_circle_property_length_equal(1,CD,C)
radius_of_circle_property_length_equal(1,CB,C)
isosceles_triangle_judgment_line_equal(1,CDB)
isosceles_triangle_property_angle_equal(1,CDB)
angle_addition(1,BCD,DCE)
Problem: As shown in the diagram, CD=x, OC=3, ∠OAC=22.5°, the center of circle O is O, AB is the diameter of circle O, CE⊥OE. Find the value of x.
问题：如图所示，CD=x，OC=3，∠OAC=22.5°，⊙O的圆心为O，圆O的直径为AB，CE垂直于OE。求x的值。
Construction (CDL):
Shape(BE,EC,OCB)
Shape(CE,EO,OC)
Shape(CO,OA,AC)
Shape(CA,OAC)
Shape(DE,EB,OBD)
Shape(AE,ED,ODA)
Collinear(BEOA)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(CD),x)
Equal(LengthOfLine(OC),3)
Equal(MeasureOfAngle(OAC),22.5)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(x)
Answer: 3*sqrt(2)
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,COA)
adjacent_complementary_angle(1,EOC,COA)
sine_theorem(1,CEO)
line_addition(1,CE,ED)
Problem: As shown in the diagram, ∠BCA=35°, the center of circle O is O. Find the measure of ∠ABO.
问题：如图所示，∠BCA=35°，圆O的圆心为O。求∠ABO的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(OD,DC,CO)
Shape(AO,OC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
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, ∠ADE=34°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠ADE=34°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(AE,EC,OCA)
Shape(CE,EO,OC)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(DA,OAD)
Shape(AD,DE,EA)
Shape(BE,ED,ODB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),34)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 112
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
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)
Problem: As shown in the diagram, the length of ⌒OCA is equal to the length of arc OAD, ∠AOC=110°, O is the center of ⊙O, DE⊥OE. Find the measure of ∠DBA.
问题：如图所示，弧OCA的长度与⌒OAD的长度相等，∠AOC=110°，圆O的圆心为O，DE垂直于OE。求∠DBA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(LengthOfArc(OCA),LengthOfArc(OAD))
Equal(MeasureOfAngle(AOC),110)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,OE)
Goal: Value(MeasureOfAngle(DBA))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OCA,O)
congruent_arc_judgment_length_equal(1,OCA,OAD)
congruent_arc_property_measure_equal(1,OCA,OAD)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠BOA=60°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=60°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BD,DA,OAB)
Shape(OD,DC,CO)
Shape(AO,OC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOC=110°, O is the center of circle O, AD∥CO. Find the measure of ∠ABD.
问题：如图所示，∠BOC=110°，⊙O的圆心为O，AD∥CO。求∠ABD的大小。
Construction (CDL):
Shape(DA,AO,OD)
Shape(AD,ODA)
Shape(DO,OB,BD)
Shape(DB,OBD)
Shape(CO,OA,OAC)
Shape(BO,OC,OCB)
Collinear(AOB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),110)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(ABD))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,BOC,COA)
parallel_property_alternate_interior_angle(1,AD,CO)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, CD=AC, ∠DCA=50°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CAB.
问题：如图所示，CD=AC，∠DCA=50°，圆O的圆心为O，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(DE,EA,AD)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(LengthOfLine(CD),LengthOfLine(AC))
Equal(MeasureOfAngle(DCA),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 25
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CAD)
isosceles_triangle_property_angle_equal(1,CAD)
triangle_property_angle_sum(1,CAD)
triangle_property_angle_sum(1,CAB)
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)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=40°，O是⊙O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(OB,BD,DO)
Shape(OD,DA,AO)
Shape(DB,BC,CD)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
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)
Problem: As shown in the diagram, AB=8, ∠COB=60°, O is the center of circle O, AB is the diameter of ⊙O. Find the length of line AC.
问题：如图所示，AB=8，∠COB=60°，⊙O的圆心为O，AB是圆O的直径。求直线AC的长度。
Construction (CDL):
Shape(CA,AO,OC)
Shape(AC,OCA)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(COB),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(AC))
Answer: 4*sqrt(3)
Theorem sequence:
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)
triangle_property_angle_sum(1,COB)
diameter_of_circle_property_right_angle(1,BCA,O)
sine_theorem(1,ABC)
Problem: As shown in the diagram, ∠COB=100°, O is the center of circle 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(OA,AD,DO)
Shape(OD,DC,CO)
Shape(AB,BD,DA)
Shape(BA,OAB)
Shape(CD,DB,OBC)
Shape(AO,OC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,CAB)
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=30°, O is the center of ⊙O, BE⊥OE. Find the measure of ∠COB.
问题：如图所示，∠ADC=30°，圆O的圆心为O，BE⊥OE。求∠COB的大小。
Construction (CDL):
Shape(AE,EC,OCA)
Shape(CE,EF,FC)
Shape(CF,FB,OBC)
Shape(AO,OE,EA)
Shape(FE,EO,OG,GF)
Shape(BF,FG,GB)
Shape(GO,OA,AD,DG)
Shape(BG,GD,ODB)
Shape(DA,OAD)
Collinear(AEFB)
Collinear(OGB)
Collinear(CEO)
Collinear(CFGD)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),30)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(MeasureOfAngle(COB))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_center_angle(1,OCA,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,BA)
perpendicular_bisector_property_bisector(1,OE,BA)
Problem: As shown in the diagram, ∠BCA=50°, ∠POA=55°, the center of circle O is O. Find the measure of ∠BOP.
问题：如图所示，∠BCA=50°，∠POA=55°，⊙O的圆心为O。求∠BOP的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(PO,OA,OAP)
Shape(BO,OP,OPB)
Shape(CB,OBC)
Cocircular(O,CAPB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
Equal(MeasureOfAngle(POA),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOP))
Answer: 45
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
angle_addition(1,BOP,POA)
Problem: As shown in the diagram, ⌒OCD is congruent to ⌒OAB, ∠BOA=40°, O is the center of circle O, AD is the diameter of ⊙O. Find the measure of ∠CPB.
问题：如图所示，弧OCD全等于弧OAB，∠BOA=40°，O是圆O的圆心，AD是⊙O的直径。求∠CPB的大小。
Construction (CDL):
Shape(PE,ED,ODP)
Shape(DE,EC,OCD)
Shape(CE,EO,OC)
Shape(CO,OB,OBC)
Shape(OF,FB,BO)
Shape(BF,FA,OAB)
Shape(EP,PF,FE)
Shape(AF,FP,OPA)
Collinear(PEC)
Collinear(DEOFA)
Collinear(PFB)
Cocircular(O,CDPAB)
Conditions (CDL):
CongruentBetweenArc(OCD,OAB)
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(CPB))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,OBC,O)
arc_property_center_angle(1,OAB,O)
congruent_arc_property_measure_equal(1,OCD,OAB)
flat_angle(1,DOA)
angle_addition(1,DOC,COB)
angle_addition(1,DCB,BOA)
arc_property_circumference_angle_external(1,OBC,P)
Problem: As shown in the diagram, ∠OBC=40°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=40°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
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, OC=6, ∠CAO=22.5°, the center of circle O is O, DE is perpendicular to OE. Find the length of line CD.
问题：如图所示，OC=6，∠CAO=22.5°，⊙O的圆心为O，DE⊥OE。求直线CD的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OE,EC)
Shape(CE,EB,OBC)
Shape(BE,ED,ODB)
Shape(DE,EA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(LengthOfLine(OC),6)
Equal(MeasureOfAngle(CAO),22.5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,OE)
Goal: Value(LengthOfLine(CD))
Answer: 6*sqrt(2)
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OE,DC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
triangle_property_angle_sum(1,COE)
isosceles_triangle_judgment_angle_equal(1,ECO)
right_triangle_judgment_angle(1,OEC)
right_triangle_property_pythagorean(1,OEC)
line_addition(1,CE,ED)
Problem: As shown in the diagram, ∠ABC=35°, the center of circle O is O, BC is the diameter of ⊙O. Find the measure of ∠BDE.
问题：如图所示，∠ABC=35°，⊙O的圆心为O，BC是⊙O的直径。求∠BDE的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(EB,BD,DE)
Shape(CE,ED,ODC)
Shape(DB,OBD)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(BDE))
Answer: 55
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OCA=20°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠CDA.
问题：如图所示，∠OCA=20°，圆O的圆心为O，圆O的直径为AB。求∠CDA的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(AD,ODA)
Shape(CD,DA,AC)
Shape(CA,AO,OC)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 110
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
triangle_property_angle_sum(1,CAO)
arc_property_circumference_angle_internal(1,OCA,D)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠BOD=130°, O is the center of circle O, AB is the diameter of circle O, AC∥DO. Find the measure of ∠OBC.
问题：如图所示，∠BOD=130°，⊙O的圆心为O，AB是圆O的直径，AC∥DO。求∠OBC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CB,OBC)
Shape(CA,AB,BC)
Shape(DO,OA,OAD)
Shape(BO,OD,ODB)
Collinear(AOB)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),130)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AC,DO)
Goal: Value(MeasureOfAngle(OBC))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,BOD,DOA)
parallel_property_alternate_interior_angle(1,AC,DO)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠BDA=70°, the center of ⊙O is O, BC is the diameter of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠BDA=70°，O是⊙O的圆心，BC是圆O的直径。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(DB,OBD)
Shape(EB,BD,DE)
Shape(CE,ED,ODC)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(BDA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 20
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAB,O)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,C)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠COB=50°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠COB=50°，圆O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(CE,EB,BC)
Shape(EO,OB,BE)
Shape(DE,EC,CD)
Shape(AO,OE,ED,DA)
Collinear(AOB)
Collinear(OEC)
Collinear(DEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 115
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_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠BOC=50°, the center of circle O is O. Find the measure of ∠BAD.
问题：如图所示，∠BOC=50°，O是⊙O的圆心。求∠BAD的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AD,DB)
Shape(BD,DC,OCB)
Shape(CD,DO,OC)
Shape(DA,AO,OD)
Shape(CO,OA,OAC)
Collinear(BDO)
Collinear(ADC)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)