SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ABC=50°, the center of circle O is O, the tangent to ⊙O is BA, AD∥BO. Find the measure of ∠DCO.
问题：如图所示，∠ABC=50°，O是⊙O的圆心，BA是圆O的切线，AD∥BO。求∠DCO的大小。
Construction (CDL):
Shape(OC,OCD,DE,EO)
Shape(OE,EC,CO)
Shape(DA,AE,ED)
Shape(EA,OAC,CE)
Shape(ODA,AD)
Shape(OAC,AB,BC)
Collinear(OCB)
Collinear(OEA)
Collinear(DEC)
Cocircular(O,ACD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
ParallelBetweenLine(AD,BO)
Goal: Value(MeasureOfAngle(DCO))
Answer: 20
Theorem sequence:
tangent_of_circle_property_perpendicular(1,BA,O,O)
triangle_property_angle_sum(1,OAB)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
parallel_property_collinear_extend(3,OB,DA,C)
parallel_property_alternate_interior_angle(2,AD,CO)
Problem: As shown in the diagram, AB=3, BC=4, O is the center of ⊙O, AG is the tangent to ⊙O, the tangent to ⊙O is BH, FK is perpendicular to OK, ABCD is a rectangle. Find the length of line EF.
问题：如图所示，AB=3，BC=4，⊙O的圆心为O，AG是圆O的切线，⊙O的切线为BH，FK垂直于OK，四边形ABCD是长方形。求直线EF的长度。
Construction (CDL):
Shape(AI,OEI,EA)
Shape(AG,OIG,IA)
Shape(GB,BH,OGH)
Shape(OG,OGH,HO)
Shape(OI,OIG,GO)
Shape(OEI,IO,OK,KE)
Shape(OFE,EK,KF)
Shape(FK,KO,OF)
Shape(OJF,FO,OJ)
Shape(OH,OHJ,JO)
Shape(OHJ,HC,CJ)
Shape(OJF,JC,CD,DF)
Collinear(AEKFD)
Collinear(AIOJC)
Collinear(AGB)
Collinear(KOH)
Collinear(BHC)
Cocircular(O,GHJFEI)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BC),4)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AG,O)
IsTangentOfCircle(BH,O)
PerpendicularBetweenLine(FK,OK)
Rectangle(ABCD)
Goal: Value(LengthOfLine(EF))
Answer: 6*sqrt(7)/7
Theorem sequence:
radius_of_circle_property_length_equal(1,OG,O)
radius_of_circle_property_length_equal(1,OH,O)
tangent_of_circle_property_perpendicular(2,AG,O,O)
tangent_of_circle_property_perpendicular(2,BH,O,O)
parallel_judgment_ipsilateral_internal_angle(1,BG,HO)
flat_angle(1,AGB)
angle_addition(1,AGO,OGB)
parallel_judgment_ipsilateral_internal_angle(1,GO,BH)
parallelogram_judgment_parallel_and_parallel(1,GBHO)
parallelogram_property_opposite_line_equal(1,GBHO)
parallel_property_collinear_extend(1,HB,OG,C)
parallel_property_corresponding_angle(2,CB,OG,A)
similar_triangle_judgment_aa(1,AGO,ABC)
line_addition(1,AG,GB)
similar_triangle_property_line_ratio(1,AGO,ABC)
similar_triangle_property_line_ratio(1,OAG,CAB)
parallel_property_collinear_extend(1,OH,GB,K)
parallel_property_collinear_extend(2,BG,HK,A)
parallel_property_collinear_extend(3,KH,AB,O)
parallel_property_collinear_extend(3,BA,OK,G)
parallel_judgment_ipsilateral_internal_angle(1,AK,GO)
parallelogram_judgment_parallel_and_parallel(1,AGOK)
parallelogram_property_opposite_line_equal(1,AGOK)
radius_of_circle_property_length_equal(1,OF,O)
right_triangle_judgment_angle(1,FKO)
right_triangle_property_pythagorean(1,FKO)
circle_property_chord_perpendicular_bisect_chord(1,O,OK,FE)
line_addition(1,EK,KF)
Problem: As shown in the diagram, ∠ABC=25°, the center of circle O is O, AB is the tangent to circle O. Find the measure of ∠CDB.
问题：如图所示，∠ABC=25°，O是圆O的圆心，圆O的切线为AB。求∠CDB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCA,AC)
Shape(CA,AE,EC)
Shape(CE,EO,OC)
Shape(CO,OB,BC)
Shape(OAD,DE,EA)
Shape(OE,ED,DB,BO)
Shape(ODB,BD)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),25)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 65
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
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, AB=sqrt(2), ∠CAB=30°, the radius of ⊙O is 1, O is the center of circle O. Find the measure of ∠ABC.
问题：如图所示，AB=sqrt(2)，∠CAB=30°，⊙O的半径为1，圆O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(OD,DC,OCA,AO)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Shape(OAB,BAD)
Shape(OBC,CB)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),sqrt(2))
Equal(MeasureOfAngle(CAB),30)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 105
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_judgment_pythagorean_inverse(1,BOA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠CBA=70°, the center of circle O is O, AP⊥DP. Find the measure of ∠DEB.
问题：如图所示，∠CBA=70°，O是⊙O的圆心，AP⊥DP。求∠DEB的大小。
Construction (CDL):
Shape(ODE,ED)
Shape(OBD,DF,FB)
Shape(DE,EF,FD)
Shape(FP,PB,BF)
Shape(OEA,AP,PF,FE)
Shape(OAC,CP,PA)
Shape(PC,CB,BP)
Shape(OCB,BC)
Collinear(DFPC)
Collinear(APB)
Collinear(EFB)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),70)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AP,DP)
Goal: Value(MeasureOfAngle(DEB))
Answer: 20
Theorem sequence:
vertical_angle(1,APD,BPC)
triangle_property_angle_sum(1,BPC)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,E)
Problem: As shown in the diagram, ∠OCA=26°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠COB.
问题：如图所示，∠OCA=26°，⊙O的圆心为O，AB是圆O的直径。求∠COB的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OC,CA,AO)
Shape(OB,OBC,CO)
Shape(OCA,AC)
Collinear(BOA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),26)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 52
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)
adjacent_complementary_angle(1,AOC,COB)
Problem: As shown in the diagram, ∠COB=60°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠OCA.
问题：如图所示，∠COB=60°，O是圆O的圆心，AB是⊙O的直径。求∠OCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(AO,OC,CA)
Shape(OAB,BO,OA)
Shape(OBC,CO,OB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OBC,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,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, ∠CAB=54°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=54°，圆O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BC)
Shape(AB,BO,OC,CA)
Shape(OBC,CO,OB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),54)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 108
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠DCE=64°, the center of ⊙O is O. Find the measure of ∠DOB.
问题：如图所示，∠DCE=64°，⊙O的圆心为O。求∠DOB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(AB,BO,OD,DA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(OB,BC,CD,DO)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),64)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 128
Theorem sequence:
flat_angle(1,BCE)
angle_addition(1,BCD,DCE)
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠AOC=90°, ∠BAC=30°, the center of ⊙O is O. Find the measure of ∠AOB.
问题：如图所示，∠AOC=90°，∠BAC=30°，O是⊙O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OCB,BC)
Shape(AD,DB,BA)
Shape(BD,DC,CB)
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Shape(OAC,CO,OA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),90)
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
angle_addition(1,AOB,BOC)
Problem: As shown in the diagram, CD=Mul(2,LengthOfLine(OE)), the center of circle O is O, the diameter of ⊙O is AB, CE⊥OE. Find the measure of ∠DCB.
问题：如图所示，CD=Mul(2,LengthOfLine(OE))，O是圆O的圆心，AB是⊙O的直径，CE垂直于OE。求∠DCB的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(OAC,CE,EO,OA)
Shape(OE,ED,DO)
Shape(EC,CB,BE)
Shape(OCB,BC)
Shape(EB,OBD,DE)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(CD),Mul(2,LengthOfLine(OE)))
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(MeasureOfAngle(DCB))
Answer: 45/2
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
line_addition(1,CE,ED)
isosceles_triangle_judgment_line_equal(1,EDO)
isosceles_triangle_property_angle_equal(1,EDO)
triangle_property_angle_sum(1,EDO)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠APB=70°, the center of circle O is O, the tangent to ⊙O is PA, PB is the tangent to circle O. Find the measure of ∠ACB.
问题：如图所示，∠APB=70°，⊙O的圆心为O，⊙O的切线为PA，⊙O的切线为PB。求∠ACB的大小。
Construction (CDL):
Shape(AP,PB,OCB,OAC)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(AC,CB,BO,OA)
Shape(AO,OD,DA)
Shape(ODA,AD)
Shape(OB,BD,DO)
Shape(OBD,DB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(APB),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 125
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
quadrilateral_property_angle_sum(1,APBO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, AB=CD, ∠BOA=42°, the center of circle O is O. Find the measure of ∠DEC.
问题：如图所示，AB=CD，∠BOA=42°，圆O的圆心为O。求∠DEC的大小。
Construction (CDL):
Shape(OF,FE,OEA,AO)
Shape(OAB,BA)
Shape(OA,AB,BO)
Shape(OB,OBC,CO)
Shape(OC,CF,FO)
Shape(FC,CD,DF)
Shape(EF,FD,DE)
Shape(ODE,ED)
Shape(OCD,DC)
Collinear(OFD)
Collinear(CFE)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(MeasureOfAngle(BOA),42)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DEC))
Answer: 21
Theorem sequence:
congruent_arc_judgment_chord_equal(1,OAB,OCD)
congruent_arc_property_measure_equal(1,OAB,OCD)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OCD,E)
Problem: As shown in the diagram, OA=3, △ABC is an equilateral △, O is the center of circle O, the diameter of ⊙O is AD. Find the length of ⌒OBD.
问题：如图所示，OA=3，三角形ABC为等边三角形，⊙O的圆心为O，圆O的直径为AD。求弧OBD的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(OA,AB,BO)
Shape(AO,OE,EC,CA)
Shape(OCA,AC)
Shape(OB,BE,EO)
Shape(OBD,DB)
Shape(BD,DE,EB)
Shape(ODC,CE,ED)
Collinear(AOED)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(OA),3)
EquilateralTriangle(ABC)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfArc(OBD))
Answer: pi
Theorem sequence:
equilateral_triangle_property_angle(1,CAB)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OBD)
isosceles_triangle_property_angle_equal(1,OBD)
triangle_property_angle_sum(1,OBD)
radius_of_circle_property_length_equal(1,OA,O)
arc_property_center_angle(1,OBD,O)
arc_length_formula(1,OBD)
Problem: As shown in the diagram, the center of circle O is O, ABCD is a square. Find the measure of ∠CPB.
问题：如图所示，O是圆O的圆心，四边形ABCD是正方形。求∠CPB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(OCD,DC)
Shape(DO,OC,CD)
Shape(OAP,PE,EA)
Shape(AE,EF,FO,OA)
Shape(OPB,BP)
Shape(PB,BE,EP)
Shape(OF,FC,CO)
Shape(EB,BF,FE)
Shape(FB,BC,CF)
Shape(OBC,CB)
Collinear(AOC)
Collinear(AEB)
Collinear(BFOD)
Collinear(PEFC)
Cocircular(O,PBCDA)
Conditions (CDL):
IsCentreOfCircle(O,O)
Square(ABCD)
Goal: Value(MeasureOfAngle(CPB))
Answer: 45
Theorem sequence:
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,P)
Problem: As shown in the diagram, ∠CAE=14°, O is the center of ⊙O, AB is the diameter of circle O, D is the midpoint of arc OCA. Find the measure of ∠BEC.
问题：如图所示，∠CAE=14°，⊙O的圆心为O，AB是⊙O的直径，D是⌒OCA的中点。求∠BEC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DC)
Shape(DA,AC,CD)
Shape(CA,AO,OB,BC)
Shape(OBC,CB)
Shape(OBC,BE,EC)
Shape(OA,OAB,BO)
Collinear(AOBE)
Collinear(DCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAE),14)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(D,OCA)
Goal: Value(MeasureOfAngle(BEC))
Answer: 24
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)
congruent_arc_judgment_length_equal(1,ODA,OCD)
congruent_arc_property_measure_equal(1,ODA,OCD)
arc_addition_measure(1,OCD,ODA)
arc_property_circumference_angle_external(1,ODA,C)
triangle_property_angle_sum(1,DAC)
angle_addition(1,DAC,CAB)
triangle_property_angle_sum(1,EDA)
Problem: As shown in the diagram, ∠ACD=40°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠DAB.
问题：如图所示，∠ACD=40°，⊙O的圆心为O，⊙O的直径为AB。求∠DAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DC)
Shape(CD,DE,EC)
Shape(DA,AE,ED)
Shape(OBC,CE,EB)
Shape(EA,AO,OB,BE)
Shape(OA,OAB,BO)
Collinear(DEB)
Collinear(AEC)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠BOA=60°, ∠OBC=55°, O is the center of circle O. Find the measure of ∠OAC.
问题：如图所示，∠BOA=60°，∠OBC=55°，圆O的圆心为O。求∠OAC的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(DB,BC,CD)
Shape(OBC,CB)
Shape(OAB,BD,DA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
Equal(MeasureOfAngle(OBC),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
triangle_property_angle_sum(1,DBC)
vertical_angle(1,ADO,CDB)
triangle_property_angle_sum(1,OAD)
Problem: As shown in the diagram, ∠BOD=140°, the center of circle O is O, AB is the diameter of circle O. Find the measure of ∠DCA.
问题：如图所示，∠BOD=140°，⊙O的圆心为O，AB是圆O的直径。求∠DCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AE,EC)
Shape(CE,EO,OB,BC)
Shape(OBC,CB)
Shape(OAD,DE,EA)
Shape(OE,ED,DO)
Shape(OD,ODB,BO)
Collinear(AEOB)
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:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
angle_addition(1,BCD,DCA)
Problem: As shown in the diagram, ∠BCD=30°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DOA.
问题：如图所示，∠BCD=30°，O是圆O的圆心，⊙O的直径为AB。求∠DOA的大小。
Construction (CDL):
Shape(OCA,AO,OE,EC)
Shape(OBC,CB)
Shape(BC,CE,EB)
Shape(OA,OAD,DO)
Shape(OD,DE,EO)
Shape(ED,ODB,BE)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DOA))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_center_angle(1,ODB,O)
flat_angle(1,BOA)
angle_addition(1,BOD,DOA)
Problem: As shown in the diagram, ∠CAB=20°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠ABC.
问题：如图所示，∠CAB=20°，O是⊙O的圆心，AB是⊙O的直径。求∠ABC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CB)
Shape(CA,AO,OB,BC)
Shape(OAB,BO,OA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 70
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, ∠BCA=38°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=38°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AO,OD,DC)
Shape(OA,AD,DO)
Shape(DA,OAB,BD)
Shape(DB,BC,CD)
Shape(OBC,CB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),38)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 76
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠CAB=40°, the center of ⊙O is O, the center of the circumscribed circle of triangle ABC is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=40°，⊙O的圆心为O，△ABC的外心为O。求∠COB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OCA,AC)
Shape(OB,BC,CO)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsCircumcenterOfTriangle(O,ABC)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠DAB=80°. Find the measure of ∠BCD.
问题：如图所示，∠DAB=80°。求∠BCD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DC)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(DA,AB,BC,CD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),80)
Goal: Value(MeasureOfAngle(BCD))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠ABC=20°, O is the center of circle O, CD is the diameter of circle O. Find the measure of ∠DCA.
问题：如图所示，∠ABC=20°，圆O的圆心为O，圆O的直径为CD。求∠DCA的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BE,EO,OD)
Shape(AD,DO,OE,EA)
Shape(AE,EC,CA)
Shape(OCA,AC)
Shape(EB,BC,CE)
Shape(OBC,CB)
Collinear(DOEC)
Collinear(BEA)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 70
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAD,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,CAD)
Problem: As shown in the diagram, ∠CAD=20°, △ABC is an equilateral △. Find the measure of ∠DCA.
问题：如图所示，∠CAD=20°，△ABC为等边△。求∠DCA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(OCA,AC)
Shape(EB,BD,DE)
Shape(CE,ED,DC)
Shape(OBD,DB)
Shape(ODC,CD)
Collinear(BEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),20)
EquilateralTriangle(ABC)
Goal: Value(MeasureOfAngle(DCA))
Answer: 100
Theorem sequence:
equilateral_triangle_property_angle(1,ABC)
angle_addition(1,CAD,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
equilateral_triangle_property_angle(1,CAB)
angle_addition(1,DCB,BCA)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of circle O, E bisects segment BC. Find the measure of ∠CBD.
问题：如图所示，∠CAB=50°，O是圆O的圆心，E平分线段BC。求∠CBD的大小。
Construction (CDL):
Shape(OAB,BF,FA)
Shape(AF,FO,OE,EC,CA)
Shape(FB,BE,EO,OF)
Shape(OCA,AC)
Shape(OBD,DB)
Shape(EB,BD,DE)
Shape(ED,DC,CE)
Shape(ODC,CD)
Collinear(AFB)
Collinear(FOED)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsMidpointOfLine(E,BC)
Goal: Value(MeasureOfAngle(CBD))
Answer: 25
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_internal(1,OBC,D)
circle_property_chord_perpendicular_bisect_chord(2,O,OE,BC)
flat_angle(1,OED)
angle_addition(1,OEC,CED)
perpendicular_bisector_judgment_per_and_mid(1,DE,CB)
perpendicular_bisector_property_distance_equal(1,DE,CB)
isosceles_triangle_judgment_line_equal(1,DCB)
isosceles_triangle_property_angle_equal(1,DCB)
triangle_property_angle_sum(1,DCB)
Problem: As shown in the diagram, ∠DCB=44°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠ABD.
问题：如图所示，∠DCB=44°，O是圆O的圆心，⊙O的直径为AB。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OE,ED)
Shape(OBD,DB)
Shape(DE,EB,BD)
Shape(OA,OAC,CE,EO)
Shape(EC,CB,BE)
Shape(OCB,BC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),44)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 46
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, BC=CD, ∠CBA=52°, O is the center of ⊙O. Find the measure of ∠BAC.
问题：如图所示，BC=CD，∠CBA=52°，圆O的圆心为O。求∠BAC的大小。
Construction (CDL):
Shape(OBA,AD,DO,OB)
Shape(OD,DB,BO)
Shape(BD,DC,CB)
Shape(OCB,BC)
Shape(DA,AC,CD)
Shape(OAC,CA)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(CD))
Equal(MeasureOfAngle(CBA),52)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 14
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,CBD)
isosceles_triangle_property_angle_equal(1,CBD)
triangle_property_angle_sum(1,CBD)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,OCB)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠FOA=40°, O is the center of ⊙O, the diameter of ⊙O is AB, E is the midpoint of ⌒OBF. Find the measure of ∠CFO.
问题：如图所示，∠FOA=40°，圆O的圆心为O，圆O的直径为AB，E是弧OBF的中点。求∠CFO的大小。
Construction (CDL):
Shape(OEA,AC,CE)
Shape(EC,CO,OE)
Shape(EO,OB,BE)
Shape(OBE,EB)
Shape(OAF,FC,CA)
Shape(CF,FO,OC)
Shape(OF,OFB,BO)
Collinear(ACOB)
Collinear(FCE)
Cocircular(O,AFBE)
Conditions (CDL):
Equal(MeasureOfAngle(FOA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(E,OBF)
Goal: Value(MeasureOfAngle(CFO))
Answer: 35
Theorem sequence:
congruent_arc_judgment_length_equal(1,OEF,OBE)
congruent_arc_property_measure_equal(1,OEF,OBE)
arc_property_center_angle(1,OEF,O)
arc_property_center_angle(1,OBE,O)
flat_angle(1,AOB)
angle_addition(1,AOE,EOB)
angle_addition(1,FOA,AOE)
radius_of_circle_property_length_equal(1,OE,O)
radius_of_circle_property_length_equal(1,OF,O)
isosceles_triangle_judgment_line_equal(1,OEF)
isosceles_triangle_property_angle_equal(1,OEF)
triangle_property_angle_sum(1,OEF)
Problem: As shown in the diagram, ∠CAB=64°, the center of ⊙O is O. Find the measure of ∠BCO.
问题：如图所示，∠CAB=64°，⊙O的圆心为O。求∠BCO的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(CA,AB,BO,OC)
Shape(CO,OB,BC)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),64)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 26
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)