SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ABC=110°. Find the measure of ∠ADE.
问题：如图所示，∠ABC=110°。求∠ADE的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(ODA,AD)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BC,CD,DA)
Shape(AD,DE)
Collinear(EDC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),110)
Goal: Value(MeasureOfAngle(ADE))
Answer: 110
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
adjacent_complementary_angle(1,CDA,ADE)
Problem: As shown in the diagram, ∠BCD=120°. Find the measure of ∠DAB.
问题：如图所示，∠BCD=120°。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(ODA,AD)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BC,CD,DA)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
Goal: Value(MeasureOfAngle(DAB))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠CDA=115°, AB is the diameter of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠CDA=115°，⊙O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(DA,AC,CD)
Shape(CA,AB,BC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),115)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 25
Theorem sequence:
arc_property_circumference_angle_internal(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,ABC)
Problem: As shown in the diagram, ∠CBE=50°, O is the center of circle O, the diameter of ⊙O is AD. Find the measure of ∠COA.
问题：如图所示，∠CBE=50°，圆O的圆心为O，⊙O的直径为AD。求∠COA的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(DO,OC,CD)
Shape(OCD,DC)
Shape(OA,AB,BC,CO)
Shape(OBC,CB)
Shape(OAB,BA)
Shape(CB,BE)
Collinear(ABE)
Collinear(AOD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CBE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 100
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBE)
arc_property_circumference_angle_internal(1,OAC,B)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠BCD=110°, O is the center of circle O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=110°，圆O的圆心为O。求∠DOB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OD,DA)
Shape(ODA,AD)
Shape(OB,BC,CD,DO)
Shape(OBC,CB)
Shape(OCD,DC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠COA=100°, O is the center of ⊙O. Find the measure of ∠ADC.
问题：如图所示，∠COA=100°，O是圆O的圆心。求∠ADC的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AO,OC,CB)
Shape(OCB,BC)
Shape(OA,AD,DC,CO)
Shape(OAD,DA)
Shape(ODC,CD)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,D)
Problem: As shown in the diagram, ∠DCE=70°. Find the measure of ∠DAB.
问题：如图所示，∠DCE=70°。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(ODA,AD)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BC,CD,DA)
Shape(DC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),70)
Goal: Value(MeasureOfAngle(DAB))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,BCD,DCE)
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,A)
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(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BC,CD,DA)
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, ∠DCE=72°, the center of circle O is O. Find the measure of ∠DOB.
问题：如图所示，∠DCE=72°，圆O的圆心为O。求∠DOB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BO,OD,DA)
Shape(OB,BC,CD,DO)
Shape(DC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 144
Theorem sequence:
adjacent_complementary_angle(1,BCD,DCE)
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠CBA=120°. Find the measure of ∠CDE.
问题：如图所示，∠CBA=120°。求∠CDE的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(OBA,AB)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(BA,AD,DC,CB)
Shape(CD,DE)
Collinear(ADE)
Cocircular(O,CBAD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),120)
Goal: Value(MeasureOfAngle(CDE))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_internal(1,OAC,D)
adjacent_complementary_angle(1,ADC,CDE)
Problem: As shown in the diagram, ∠AOB=70°, the center of circle O is O, BE⊥OE. Find the measure of ∠CDA.
问题：如图所示，∠AOB=70°，O是⊙O的圆心，BE⊥OE。求∠CDA的大小。
Construction (CDL):
Shape(ODB,BO,OE,EF,FD)
Shape(DF,FC,CD)
Shape(OCD,DC)
Shape(OB,BE,EO)
Shape(OBA,AE,EB)
Shape(EA,AF,FE)
Shape(CF,FA,OAC)
Collinear(BEFC)
Collinear(OEA)
Collinear(DFA)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),70)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(MeasureOfAngle(CDA))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OBA,O)
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)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, Add(MeasureOfAngle(DEF)=MeasureOfAngle(FDE)). Find the measure of ∠CFE.
问题：如图所示，Add(MeasureOfAngle(DEF)=MeasureOfAngle(FDE))。求∠CFE的大小。
Construction (CDL):
Shape(CF,FE)
Shape(EF,FD,DE)
Collinear(CFD)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DEF),MeasureOfAngle(FDE)),45)
Goal: Value(MeasureOfAngle(CFE))
Answer: 45
Theorem sequence:
triangle_property_angle_sum(1,EFD)
adjacent_complementary_angle(1,CFE,EFD)
Problem: As shown in the diagram, ∠OBC=26°, O is the center of circle O, the diameter of ⊙O is AC. Find the measure of ∠BOA.
问题：如图所示，∠OBC=26°，⊙O的圆心为O，圆O的直径为AC。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(CO,OB,BC)
Shape(OAB,BO,OA)
Shape(OBC,CB)
Collinear(AOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),26)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 52
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)
adjacent_complementary_angle(1,COB,BOA)
Problem: As shown in the diagram, ∠DAB=108°, CF is the angle bisector of ∠DCE. Find the measure of ∠DCF.
问题：如图所示，∠DAB=108°，CF是∠DCE的角平分线。求∠DCF的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BC,CD,DA)
Shape(DC,CF)
Shape(FC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),108)
IsBisectorOfAngle(CF,DCE)
Goal: Value(MeasureOfAngle(DCF))
Answer: 54
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
adjacent_complementary_angle(1,BCD,DCE)
angle_addition(1,DCF,FCE)
Problem: As shown in the diagram, AC=2, ∠BMC=120°, AB is the diameter of circle O. Find the radius of ⊙O.
问题：如图所示，AC=2，∠BMC=120°，⊙O的直径为AB。求圆O的半径。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BC,CA)
Shape(OCA,AC)
Shape(OBM,MB)
Shape(CB,BM,MC)
Shape(OMC,CM)
Cocircular(O,BMCA)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(MeasureOfAngle(BMC),120)
IsDiameterOfCircle(AB,O)
Goal: Value(RadiusOfCircle(O))
Answer: 2
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_internal(1,OBC,M)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,ABC)
sine_theorem(1,ABC)
diameter_of_circle_property_length_equal(1,AB,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, ∠BCA=70°, the center of ⊙O is O. Find the measure of ∠ADB.
问题：如图所示，∠BCA=70°，O是圆O的圆心。求∠ADB的大小。
Construction (CDL):
Shape(PAD,DA)
Shape(PDB,BD)
Shape(AD,DB,OAB)
Shape(OAB,BO,OA)
Shape(POA,AO)
Shape(PBO,OB)
Shape(CA,POA,PBO,BC)
Shape(OCA,AC)
Shape(OBC,CB)
Cocircular(P,DBOA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADB))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,PBA,O)
arc_property_circumference_angle_external(1,PBA,D)
Problem: As shown in the diagram, ∠DBA=55°, the diameter of circle O is AB. Find the measure of ∠BCD.
问题：如图所示，∠DBA=55°，AB是⊙O的直径。求∠BCD的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAD,DA)
Shape(ODB,BD)
Shape(OBC,CB)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(AD,DE,EA)
Shape(ED,DB,BE)
Collinear(AEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),55)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 35
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, ∠DAB=100°. Find the measure of arc ODB.
问题：如图所示，∠DAB=100°。求弧ODB的角度。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(DA,AB,OBD)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),100)
Goal: Value(MeasureOfArc(ODB))
Answer: 160
Theorem sequence:
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, ∠CBA=75°, ∠DCB=85°. Find the measure of ∠ADC minus the measure of ∠BAD.
问题：如图所示，∠CBA=75°，∠DCB=85°。求∠ADC的大小减去∠BAD的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(AD,DC,CB,BA)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),75)
Equal(MeasureOfAngle(DCB),85)
Goal: Value(Sub(MeasureOfAngle(ADC),MeasureOfAngle(BAD)))
Answer: 10
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_internal(1,OAC,D)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_internal(1,OBD,A)
Problem: As shown in the diagram, the measure of arc OBD is 240. Find the measure of ∠BCD.
问题：如图所示，⌒OBD的角度为240。求∠BCD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(DA,AB,BC,CD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfArc(OBD),240)
Goal: Value(MeasureOfAngle(BCD))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DAB=50°. Find the measure of ∠BCD.
问题：如图所示，∠DAB=50°。求∠BCD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),50)
Goal: Value(MeasureOfAngle(BCD))
Answer: 130
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DOB=110°, the center of circle O is O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=110°，O是圆O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AB,BO,OD)
Shape(DO,OB,BC,CD)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠ABC=50°. Find the measure of ∠CDA.
问题：如图所示，∠ABC=50°。求∠CDA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(AB,BC,CD,DA)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
Goal: Value(MeasureOfAngle(CDA))
Answer: 130
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠DOB=138°, the center of ⊙O is O. Find the measure of ∠DCE.
问题：如图所示，∠DOB=138°，O是圆O的圆心。求∠DCE的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OD,DA)
Shape(OB,BC,CD,DO)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(OCD,CE,ED)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),138)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCE))
Answer: 69
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
adjacent_complementary_angle(1,BCD,DCE)
Problem: As shown in the diagram, ∠OEA=40°, O is the center of circle O, EA is perpendicular to OA. Find the measure of ∠FCD.
问题：如图所示，∠OEA=40°，O是圆O的圆心，EA垂直于OA。求∠FCD的大小。
Construction (CDL):
Shape(EO,OC,OCE)
Shape(OE,EA,AO)
Shape(AE,OED,DA)
Shape(AD,ODF,FA)
Shape(CA,AF,FC)
Shape(CF,OFC)
Collinear(COAD)
Collinear(EAF)
Cocircular(O,EDFC)
Conditions (CDL):
Equal(MeasureOfAngle(OEA),40)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(EA,OA)
Goal: Value(MeasureOfAngle(FCD))
Answer: 25
Theorem sequence:
triangle_property_angle_sum(1,OEA)
arc_property_center_angle(1,OED,O)
circle_property_chord_perpendicular_bisect_arc(1,OEF,OAD)
congruent_arc_judgment_length_equal(1,OED,ODF)
congruent_arc_property_measure_equal(1,OED,ODF)
arc_property_circumference_angle_external(1,ODF,C)
Problem: As shown in the diagram, ∠BEC=36°, ∠CFD=30°, O is the center of ⊙O. Find the measure of ∠DAE.
问题：如图所示，∠BEC=36°，∠CFD=30°，⊙O的圆心为O。求∠DAE的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AB,BC,CD,DA)
Shape(DC,OCD)
Shape(OBC,BE,EC)
Shape(CF,FD,OCD)
Collinear(ABE)
Collinear(BCF)
Collinear(DCE)
Collinear(ADF)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BEC),36)
Equal(MeasureOfAngle(CFD),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAE))
Answer: 57
Theorem sequence:
triangle_property_angle_sum(1,DAE)
triangle_property_angle_sum(1,ABF)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_internal(1,OAC,B)
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(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(AB,BC,CD,DA)
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, ∠BDA=100°, E is the center of circle E, the center of ⊙O is O. Find the measure of ∠ACB.
问题：如图所示，∠BDA=100°，E是⊙E的圆心，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CB,EAB)
Shape(BC,OCB)
Shape(AO,EAO)
Shape(AO,OB,OBA)
Shape(BO,EOB)
Shape(DB,EBD)
Shape(AD,EDA)
Shape(BD,DA,OBA)
Cocircular(O,CBA)
Cocircular(E,AOBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDA),100)
IsCentreOfCircle(E,E)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_circumference_angle_external(1,EAB,D)
arc_property_circumference_angle_internal(1,EAB,O)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠ABC=30°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠ABC=30°，O是圆O的圆心。求∠CDA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CD,DA)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(CB,OBC)
Collinear(BOC)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 150
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠BOA=120°, O is the center of circle O. Find the measure of ∠BPC.
问题：如图所示，∠BOA=120°，O是圆O的圆心。求∠BPC的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,AP,PB,BO)
Shape(PA,OAP)
Shape(BP,OPB)
Shape(PC,CB,OPB)
Collinear(APC)
Cocircular(O,APB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BPC))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,P)
adjacent_complementary_angle(1,APB,BPC)
Problem: As shown in the diagram, ∠ABO=20°, ∠CAO=40°, the center of circle O is O. Find the measure of ∠OBC.
问题：如图所示，∠ABO=20°，∠CAO=40°，⊙O的圆心为O。求∠OBC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(BO,OA,AB)
Shape(OB,BC,CO)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),20)
Equal(MeasureOfAngle(CAO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 30
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)
triangle_property_angle_sum(1,AOC)
isosceles_triangle_property_angle_equal(1,OCA)
angle_addition(1,ABO,OBC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠CDB=30°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠CDB=30°，O是圆O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BE,ED)
Shape(EB,OBC,CE)
Shape(EC,CO,OE)
Shape(DE,EA,OAD)
Shape(AO,OC,OCA)
Collinear(DEC)
Collinear(BEOA)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
adjacent_complementary_angle(1,AOC,COB)
Problem: As shown in the diagram, ∠CAB=26°, the center of ⊙O is O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=26°，O是⊙O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(BO,OE,EA,OAB)
Shape(OB,BE,EO)
Shape(EB,BC,CE)
Shape(EC,CA,AE)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(OEC)
Collinear(AEB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),26)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
triangle_property_angle_sum(1,OBC)
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)
Problem: As shown in the diagram, AC=3, ∠ABC=30°, ∠BCA=60°. Find the radius of circle O.
问题：如图所示，AC=3，∠ABC=30°，∠BCA=60°。求圆O的半径。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(MeasureOfAngle(ABC),30)
Equal(MeasureOfAngle(BCA),60)
Goal: Value(RadiusOfCircle(O))
Answer: 3
Theorem sequence:
triangle_property_angle_sum(1,ABC)
sine_theorem(1,CAB)
diameter_of_circle_judgment_right_angle(1,CAB,O)
diameter_of_circle_property_length_equal(1,BC,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, ∠COB=120°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COB=120°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, AB=6, ∠ACB=30°, O is the center of circle O. Find the radius of circle O.
问题：如图所示，AB=6，∠ACB=30°，O是圆O的圆心。求⊙O的半径。
Construction (CDL):
Shape(OE,EC,OCB,BO)
Shape(CE,EA,AC)
Shape(CA,OAC)
Shape(AB,OBA)
Shape(OB,BE,EO)
Shape(EB,BA,AE)
Collinear(CEB)
Collinear(OEA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(MeasureOfAngle(ACB),30)
IsCentreOfCircle(O,O)
Goal: Value(RadiusOfCircle(O))
Answer: 6
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OBA)
triangle_property_angle_sum(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
sine_theorem(1,AOB)
Problem: As shown in the diagram, ∠CAB=20°, the center of circle O is O, DE is perpendicular to OE. Find the measure of ∠BOD.
问题：如图所示，∠CAB=20°，圆O的圆心为O，DE⊥OE。求∠BOD的大小。
Construction (CDL):
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(CE,EB,OBC)
Shape(OD,DE,EO)
Shape(ED,ODB,BE)
Shape(OA,OAD,DO)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,OE)
Goal: Value(MeasureOfAngle(BOD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,ODB,O)
circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)
congruent_arc_judgment_length_equal(1,ODB,OBC)
congruent_arc_property_measure_equal(1,ODB,OBC)
Problem: As shown in the diagram, ∠ADC=115°, the center of ⊙O is O. Find the measure of ∠COA.
问题：如图所示，∠ADC=115°，O是⊙O的圆心。求∠COA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OC,CB)
Shape(BC,OCB)
Shape(OA,AC,CO)
Shape(AD,DC,CA)
Shape(DA,OAD)
Shape(CD,ODC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),115)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 130
Theorem sequence:
arc_property_circumference_angle_internal(1,OAC,D)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, the length of arc OBC is equal to the length of ⌒OCA, ∠BCA=100°. Find the measure of ∠ADC.
问题：如图所示，⌒OBC的长度与⌒OCA的长度相等，∠BCA=100°。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(EA,AD,DE)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(CE,EB,BC)
Shape(CB,OBC)
Collinear(CED)
Collinear(AEB)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OCA))
Equal(MeasureOfAngle(BCA),100)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OCA)
congruent_arc_property_chord_equal(1,OBC,OCA)
triangle_property_angle_sum(1,CAB)
isosceles_triangle_judgment_line_equal(1,CAB)
isosceles_triangle_property_angle_equal(1,CAB)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠DCA=40°, AB is the diameter of circle O. Find the measure of ∠BAD.
问题：如图所示，∠DCA=40°，圆O的直径为AB。求∠BAD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(EA,AD,DE)
Shape(DA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Shape(CE,EB,BC)
Shape(CB,OBC)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),40)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
Problem: As shown in the diagram, ∠COA=160°, O is the center of circle O. Find the measure of ∠ADC.
问题：如图所示，∠COA=160°，⊙O的圆心为O。求∠ADC的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OC,CB)
Shape(BC,OCB)
Shape(OA,AC,CO)
Shape(DA,OAD)
Shape(AD,DC,CA)
Shape(CD,ODC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),160)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,D)
Problem: As shown in the diagram, ∠OCA=38°, O is the center of circle O. Find the measure of ∠ABC.
问题：如图所示，∠OCA=38°，圆O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CO,OA)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),38)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
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,AOC)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)