SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCD=25°, ∠DEA=39°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠BCD=25°，∠DEA=39°，⊙O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(AF,FB,BA)
Shape(OCD,DC)
Shape(OBA,AB)
Shape(BF,FD,ODB)
Shape(FC,CD,DF)
Shape(ODB,DE,EB)
Collinear(ABE)
Collinear(CDE)
Collinear(CFB)
Collinear(AFD)
Cocircular(O,CDBA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),25)
Equal(MeasureOfAngle(DEA),39)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_external(1,ODB,A)
triangle_property_angle_sum(1,ADE)
flat_angle(1,CDE)
angle_addition(1,CDF,FDE)
Problem: As shown in the diagram, AB=AD, ∠DOC=120°, the center of ⊙O is O, DB is the diameter of circle O. Find the measure of ∠BGA.
问题：如图所示，AB=AD，∠DOC=120°，圆O的圆心为O，圆O的直径为DB。求∠BGA的大小。
Construction (CDL):
Shape(DA,AG,GO,OD)
Shape(CO,OG,GC)
Shape(GA,AB,BG)
Shape(ODA,AD)
Shape(DO,OC,OCD)
Collinear(DOGB)
Collinear(AGC)
Cocircular(O,DABC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(MeasureOfAngle(DOC),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DB,O)
Goal: Value(MeasureOfAngle(BGA))
Answer: 105
Theorem sequence:
flat_angle(1,DOG)
angle_addition(1,DOC,COG)
arc_property_center_angle(1,OBD,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBC,A)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,GAB)
Problem: As shown in the diagram, ∠APO=45°, O is the center of ⊙O, PA is the tangent to circle O. Find the measure of ∠BCA.
问题：如图所示，∠APO=45°，O是⊙O的圆心，圆O的切线为PA。求∠BCA的大小。
Construction (CDL):
Shape(OD,DB,BO)
Shape(BD,DA,OAB)
Shape(OAB,AP,PB)
Shape(DC,CA,AD)
Collinear(CDB)
Collinear(ODA)
Collinear(OBP)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(APO),45)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 45/2
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
triangle_property_angle_sum(1,OAP)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ABC=60°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=60°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(AO,OC,CA)
Shape(AB,BC,CO,OA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠OBA=25°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠OBA=25°，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(AC,CB,BO,OA)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OBA,AB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 65
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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BCA=50°, O is the center of circle O, the diameter of ⊙O is AD. Find the measure of ∠DAB.
问题：如图所示，∠BCA=50°，⊙O的圆心为O，AD是⊙O的直径。求∠DAB的大小。
Construction (CDL):
Shape(CA,AE,EC)
Shape(EA,AB,BE)
Shape(EB,BD,DO,OE)
Collinear(AEOD)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABD,O)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,C)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, ∠CAD=50°, the center of circle O is O, the diameter of ⊙O is AD. Find the measure of ∠ABC.
问题：如图所示，∠CAD=50°，O是圆O的圆心，圆O的直径为AD。求∠ABC的大小。
Construction (CDL):
Shape(ED,DC,CE)
Shape(OE,EC,CA,AO)
Shape(BE,EO,OA,AB)
Collinear(BEC)
Collinear(DEOA)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OAD,C)
triangle_property_angle_sum(1,DCA)
flat_angle(1,DOA)
Problem: As shown in the diagram, ∠BEO=90°, ∠CDB=70°, O is the center of circle O. Find the measure of ∠OAC.
问题：如图所示，∠BEO=90°，∠CDB=70°，O是⊙O的圆心。求∠OAC的大小。
Construction (CDL):
Shape(DB,BO,OC,CD)
Shape(OB,BE,EO)
Shape(OE,EC,CO)
Shape(CE,EA,AC)
Collinear(BEC)
Collinear(OEA)
Cocircular(O,DBAC)
Conditions (CDL):
Equal(MeasureOfAngle(BEO),90)
Equal(MeasureOfAngle(CDB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_center_angle(1,OBC,O)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OBC,D)
circle_property_chord_perpendicular_bisect_arc(1,OBC,OEA)
congruent_arc_judgment_length_equal(1,OBA,OAC)
congruent_arc_property_measure_equal(1,OBA,OAC)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,OBE)
triangle_property_angle_sum(1,OBC)
triangle_property_angle_sum(1,COA)
Problem: As shown in the diagram, ∠ABD=54°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DCB.
问题：如图所示，∠ABD=54°，O是圆O的圆心，AB是⊙O的直径。求∠DCB的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EO,OB,BD)
Shape(BO,OE,EC,CB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),54)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 36
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠ABO=40°, ∠COA=120°, ∠OAB=90°. Find the measure of ∠COB.
问题：如图所示，∠ABO=40°，∠COA=120°，∠OAB=90°。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(CO,OB)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),40)
Equal(MeasureOfAngle(COA),120)
Equal(MeasureOfAngle(OAB),90)
Goal: Value(MeasureOfAngle(COB))
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,ABO)
angle_addition(1,COB,BOA)
Problem: As shown in the diagram, ∠OBC=35°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=35°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
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, ∠OBC=25°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=25°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 65
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
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, OA=2, ∠BCA=45°, the center of circle O is O. Find the length of line AB.
问题：如图所示，OA=2，∠BCA=45°，⊙O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Cocircular(O,CAB)
Conditions (CDL):
Equal(LengthOfLine(OA),2)
Equal(MeasureOfAngle(BCA),45)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(2)
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)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, ∠CAB=74°, O is the center of circle O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=74°，O是⊙O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),74)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 16
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
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, ∠DBA=20°, ∠DEA=80°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠DBA=20°，∠DEA=80°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OCA,AE,EC)
Shape(CE,EO,OC)
Shape(OAD,DE,EA)
Shape(ED,DB,BO,OE)
Shape(ODB,BD)
Shape(OBC,CO,OB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),20)
Equal(MeasureOfAngle(DEA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 120
Theorem sequence:
adjacent_complementary_angle(1,BED,DEA)
triangle_property_angle_sum(1,DBE)
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠BOA=70°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=70°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OD,DC,OCA,AO)
Shape(OA,AD,DO)
Shape(DA,OAB,BD)
Shape(DB,BC,CD)
Shape(OBC,CB)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 35
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BCA.
问题：如图所示，O是⊙O的圆心，⊙O的直径为AB。求∠BCA的大小。
Construction (CDL):
Shape(OB,BC,CA,AO)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OA,OAB,BO)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 90
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
Problem: As shown in the diagram, ∠ADC=25°, the center of circle O is O, BA is the tangent to circle O. Find the measure of ∠OBA.
问题：如图所示，∠ADC=25°，O是圆O的圆心，BA是⊙O的切线。求∠OBA的大小。
Construction (CDL):
Shape(OE,ED,ODC,CO)
Shape(OC,CE,EO)
Shape(EA,AD,DE)
Shape(EC,OCA,AE)
Shape(OAD,DA)
Shape(CB,BA,OCA)
Collinear(BCO)
Collinear(CED)
Collinear(OEA)
Cocircular(O,CAD)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),25)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(OBA))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_center_angle(1,OCA,O)
tangent_of_circle_property_perpendicular(2,BA,O,O)
triangle_property_angle_sum(1,OBA)
Problem: As shown in the diagram, ∠CBE=56°, the center of circle O is O. Find the measure of ∠COA.
问题：如图所示，∠CBE=56°，O是圆O的圆心。求∠COA的大小。
Construction (CDL):
Shape(DA,AO,OC,CD)
Shape(ODA,AD)
Shape(OCD,DC)
Shape(OA,AB,BC,CO)
Shape(OAB,BA)
Shape(OBC,CB)
Collinear(ABE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CBE),56)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 112
Theorem sequence:
arc_property_circumference_angle_internal(1,OAC,B)
adjacent_complementary_angle(1,ABC,CBE)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠CAB=50°, O is the incenter of triangle ABC. Find the measure of ∠COB.
问题：如图所示，∠CAB=50°，△ABC的内切圆圆心为O。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Shape(OB,BC,CO)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsIncenterOfTriangle(O,ABC)
Goal: Value(MeasureOfAngle(COB))
Answer: 115
Theorem sequence:
angle_addition(1,ABO,OBC)
angle_addition(1,BCO,OCA)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠DBA=63°, ∠DCO=24°, O is the center of ⊙O, BA is the diameter of ⊙O. Find the measure of ∠CDB.
问题：如图所示，∠DBA=63°，∠DCO=24°，⊙O的圆心为O，⊙O的直径为BA。求∠CDB的大小。
Construction (CDL):
Shape(ODB,BD)
Shape(DB,BE,ED)
Shape(DE,EO,OA,OAD)
Shape(OBC,CE,EB)
Shape(EC,CO,OE)
Shape(OC,CA,AO)
Shape(OCA,AC)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),63)
Equal(MeasureOfAngle(DCO),24)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 39
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
angle_addition(1,DCO,OCA)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BOC=56°, O is the center of circle O, C is the midpoint of segment AB. Find the measure of ∠ADB.
问题：如图所示，∠BOC=56°，圆O的圆心为O，C是线段AB的中点。求∠ADB的大小。
Construction (CDL):
Shape(OPA,AP)
Shape(OBP,PB)
Shape(PA,AO,OB,BP)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(CA,AD,DB,BC)
Shape(OAD,DA)
Shape(ODB,BD)
Collinear(ACB)
Cocircular(O,ADBP)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),56)
IsCentreOfCircle(O,O)
IsMidpointOfLine(C,AB)
Goal: Value(MeasureOfAngle(ADB))
Answer: 124
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(2,O,OC,AB)
perpendicular_bisector_property_bisector(1,OC,AB)
angle_addition(1,BOC,COA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,D)
Problem: As shown in the diagram, ∠CDA=120°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠CEB.
问题：如图所示，∠CDA=120°，O是⊙O的圆心，AB是圆O的直径。求∠CEB的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(OBC,CD)
Shape(ODE,EF,FD)
Shape(OEA,AG,GE)
Shape(EG,GF,FE)
Shape(FG,GH,HC,CF)
Shape(DF,FC,CD)
Shape(GA,AH,HG)
Shape(CH,HB,BC)
Shape(HA,AO,OB,BH)
Shape(OAB,BO,OA)
Collinear(AGFD)
Collinear(EGHB)
Collinear(EFC)
Collinear(AHC)
Collinear(AOB)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CEB))
Answer: 30
Theorem sequence:
round_arc(1,OCA,OAC)
arc_property_circumference_angle_internal(1,OCA,D)
arc_property_circumference_angle_internal(1,OAC,B)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,E)
Problem: As shown in the diagram, ∠ABC=25°, the center of circle O is O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=25°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OCA,AC)
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Shape(DO,OC,CD)
Shape(OD,DB,OBC,CO)
Collinear(ADO)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠BCA=43°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=43°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OBC,CB)
Shape(OA,OAB,BO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),43)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 86
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠AOC=60°, O is the center of circle O, BC is the diameter of circle O. Find the measure of ∠OAB.
问题：如图所示，∠AOC=60°，⊙O的圆心为O，圆O的直径为BC。求∠OAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(BO,OA,AB)
Shape(OBC,CO,OB)
Shape(OC,OCA,AO)
Collinear(BOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
Problem: As shown in the diagram, the center of ⊙O is O, the diameter of ⊙O is AB, D bisects arc OAB. Find the measure of ∠BCD.
问题：如图所示，⊙O的圆心为O，圆O的直径为AB，D平分⌒OAB。求∠BCD的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(OBC,CB)
Shape(CE,EB,BC)
Shape(OA,OAD,DO)
Shape(OD,DE,EO)
Shape(ED,ODB,BE)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(D,OAB)
Goal: Value(MeasureOfAngle(BCD))
Answer: 45
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAD,ODB)
congruent_arc_property_measure_equal(1,OAD,ODB)
arc_property_center_angle(1,OAD,O)
arc_property_center_angle(1,ODB,O)
adjacent_complementary_angle(1,BOD,DOA)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠CDA=27°, O is the center of circle O, BA is the tangent to circle O. Find the measure of ∠ABC.
问题：如图所示，∠CDA=27°，O是⊙O的圆心，BA是⊙O的切线。求∠ABC的大小。
Construction (CDL):
Shape(OC,OCD,DO)
Shape(OD,DE,EO)
Shape(OE,EC,CO)
Shape(DA,AE,ED)
Shape(EA,OAC,CE)
Shape(ODA,AD)
Shape(OAC,AB,BC)
Collinear(DEC)
Collinear(OEA)
Collinear(OCB)
Cocircular(O,ACD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),27)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 36
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
arc_property_center_angle(1,OAC,O)
tangent_of_circle_property_perpendicular(1,BA,O,O)
triangle_property_angle_sum(1,OAB)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of ⊙O, BE⊥OE. Find the measure of ∠CBD.
问题：如图所示，∠CAB=50°，O是圆O的圆心，BE⊥OE。求∠CBD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OCA,AC)
Shape(AB,BE,EO,OA)
Shape(AO,OE,EC,CA)
Shape(BD,DE,EB)
Shape(ED,DC,CE)
Shape(OBD,DB)
Shape(ODC,CD)
Collinear(BEC)
Collinear(OED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(MeasureOfAngle(CBD))
Answer: 25
Theorem sequence:
arc_property_circumference_angle_internal(1,OBC,D)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,BC)
vertical_angle(1,BEO,CED)
perpendicular_bisector_judgment_per_and_mid(1,DE,CB)
perpendicular_bisector_property_distance_equal(1,DE,CB)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,DCB)
isosceles_triangle_judgment_line_equal(1,DCB)
isosceles_triangle_property_angle_equal(1,DCB)
Problem: As shown in the diagram, ∠ACB=70°, the center of ⊙O is O. Find the measure of ∠BAO.
问题：如图所示，∠ACB=70°，⊙O的圆心为O。求∠BAO的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OCB,BC)
Shape(BA,AD,DB)
Shape(BD,DC,CB)
Shape(DA,AO,OD)
Shape(OA,OAC,CD,DO)
Collinear(ADC)
Collinear(BDO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAO))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,OBA)
Problem: As shown in the diagram, ∠ACE=20°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BDE.
问题：如图所示，∠ACE=20°，⊙O的圆心为O，圆O的直径为AB。求∠BDE的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OEA,AE)
Shape(ODE,ED)
Shape(OBD,DB)
Shape(AC,CF,FA)
Shape(AF,FG,GA)
Shape(AG,GE,EA)
Shape(FO,OB,BD,DG,GF)
Shape(GD,DE,EG)
Shape(OF,FC,OCB,BO)
Collinear(CFGE)
Collinear(AFOB)
Collinear(AGD)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDE))
Answer: 110
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,OEA,C)
arc_property_circumference_angle_external(1,OEA,D)
angle_addition(1,BDA,ADE)
Problem: As shown in the diagram, ∠CAB=35°, the center of ⊙O is O, AB is the diameter of ⊙O, the tangent to circle O is CB. Find the measure of ∠BCA.
问题：如图所示，∠CAB=35°，⊙O的圆心为O，⊙O的直径为AB，CB是圆O的切线。求∠BCA的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(ODA,AD)
Shape(AO,OB,OBD,DA)
Shape(OBD,BC,CD)
Collinear(ADC)
Collinear(AOB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CB,O,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠DCB=Mul(2,MeasureOfAngle(BAD))°, O is the center of ⊙O, the diameter of ⊙O is BE. Find the measure of ∠DAE.
问题：如图所示，∠DCB=Mul(2,MeasureOfAngle(BAD))°，O是⊙O的圆心，BE是圆O的直径。求∠DAE的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OCB,BC)
Shape(BA,AF,FO,OB)
Shape(BO,OF,FD,DC,CB)
Shape(ODC,CD)
Shape(FA,AE,EF)
Shape(FE,OED,DF)
Shape(OAE,EA)
Collinear(AFD)
Collinear(BOFE)
Cocircular(O,AEDCB)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),Mul(2,MeasureOfAngle(BAD)))
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Goal: Value(MeasureOfAngle(DAE))
Answer: 30
Theorem sequence:
round_arc(1,ODB,OBD)
arc_property_circumference_angle_internal(1,ODB,C)
arc_property_circumference_angle_internal(1,OBD,A)
diameter_of_circle_property_right_angle(1,BAE,O)
angle_addition(1,BAD,DAE)
Problem: As shown in the diagram, ∠COA=50°, O is the center of ⊙O, AC is the tangent to circle O. Find the measure of ∠ACD.
问题：如图所示，∠COA=50°，⊙O的圆心为O，圆O的切线为AC。求∠ACD的大小。
Construction (CDL):
Shape(OC,OCD,DO)
Shape(OD,DC,CO)
Shape(ODC,CD)
Shape(DA,AC,ODC)
Collinear(ADO)
Cocircular(O,DC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AC,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AC,O,O)
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,ACD,DCO)
Problem: As shown in the diagram, ∠BPA=70°, O is the center of circle O, PA is the tangent to circle O, PB is the tangent to ⊙O. Find the measure of ∠OBA.
问题：如图所示，∠BPA=70°，O是⊙O的圆心，圆O的切线为PA，PB是⊙O的切线。求∠OBA的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(OB,BA,AO)
Shape(OBA,AB)
Shape(BP,PA,OBA)
Cocircular(O,BA)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(OBA))
Answer: 35
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PB,O,O)
tangent_of_circle_property_perpendicular(2,PA,O,O)
quadrilateral_property_angle_sum(1,OBPA)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,OBA)