SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠AOB=50°, O is the center of circle O, BQ is perpendicular to OQ. Find the measure of ∠CDA.
问题：如图所示，∠AOB=50°，⊙O的圆心为O，BQ⊥OQ。求∠CDA的大小。
Construction (CDL):
Shape(OAC,CP,PA)
Shape(AP,PQ,QA)
Shape(AQ,QB,OBA)
Shape(PC,CM,MP)
Shape(PM,MO,OQ,QP)
Shape(QO,OB,BQ)
Shape(CD,DM,MC)
Shape(OCD,DC)
Shape(OM,MD,ODB,BO)
Collinear(APMD)
Collinear(AQO)
Collinear(CPQB)
Collinear(CMO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BQ,OQ)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
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)
altitude_of_triangle_judgment(1,OQ,OBC)
isosceles_triangle_property_line_coincidence(1,OBC,Q)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠CAO=20°, ∠CBO=50°, the center of circle O is O. Find the measure of ∠AOB.
问题：如图所示，∠CAO=20°，∠CBO=50°，⊙O的圆心为O。求∠AOB的大小。
Construction (CDL):
Shape(OA,OAC,CM,MO)
Shape(AO,OM,MA)
Shape(MC,CB,BM)
Shape(AM,MB,OBA)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),20)
Equal(MeasureOfAngle(CBO),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
vertical_angle(1,OMA,BMC)
triangle_property_angle_sum(1,AOM)
triangle_property_angle_sum(1,MCB)
Problem: As shown in the diagram, ∠ABO=40°, ∠ADC=110°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠ABO=40°，∠ADC=110°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BD,DA)
Shape(AD,DO,OC,CA)
Shape(DB,BO,OD)
Shape(OB,OBC,CO)
Shape(AC,OCA)
Collinear(ADB)
Collinear(COD)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),40)
Equal(MeasureOfAngle(ADC),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,ODB)
adjacent_complementary_angle(1,ADO,ODB)
adjacent_complementary_angle(1,COB,BOD)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, AB=4, BC=3, O is the center of circle O. Find the diameter of ⊙O.
问题：如图所示，AB=4，BC=3，O是圆O的圆心。求圆O的直径。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AO,OC,CB)
Shape(CO,OA,OAC)
Shape(BC,OCB)
Collinear(AOC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),3)
IsCentreOfCircle(O,O)
Goal: Value(DiameterOfCircle(O))
Answer: 5
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOC,O)
diameter_of_circle_property_right_angle(1,CBA,O)
right_triangle_judgment_angle(1,CBA)
right_triangle_property_pythagorean(1,CBA)
diameter_of_circle_property_length_equal(1,AC,O)
Problem: As shown in the diagram, ∠CBA=30°, the radius of circle O is 2, O is the center of circle O, the diameter of ⊙O is AB, CD⊥AD. Find the length of line CD.
问题：如图所示，∠CBA=30°，⊙O的半径为2，O是⊙O的圆心，⊙O的直径为AB，CD⊥AD。求直线CD的长度。
Construction (CDL):
Shape(OB,OBA,AD,DO)
Shape(BO,OD,DC,CB)
Shape(BC,OCB)
Shape(DA,AC,CD)
Shape(CA,OAC)
Collinear(ADOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),30)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CD,AD)
Goal: Value(LengthOfLine(CD))
Answer: sqrt(3)
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
diameter_of_circle_property_length_equal(1,AB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
sine_theorem(1,ACB)
sine_theorem(1,CDA)
Problem: As shown in the diagram, ∠BDC=40°, the center of circle O is O, the diameter of circle O is AC. Find the measure of ∠ACB.
问题：如图所示，∠BDC=40°，O是圆O的圆心，AC是⊙O的直径。求∠ACB的大小。
Construction (CDL):
Shape(OA,OAD,DM,MO)
Shape(AO,OM,MB,BA)
Shape(AB,OBA)
Shape(MD,DC,CM)
Shape(CD,ODC)
Shape(MC,CB,BM)
Shape(BC,OCB)
Collinear(DMB)
Collinear(CMOA)
Cocircular(O,DCBA)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,D)
arc_property_circumference_angle_external(1,OCB,A)
diameter_of_circle_property_right_angle(1,CBA,O)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, ∠COB=80°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=80°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,OBC,CO)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOD=100°, O is the center of ⊙O. Find the measure of ∠BCD.
问题：如图所示，∠BOD=100°，O是⊙O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OD,DA)
Shape(AD,ODA)
Shape(OB,BC,CD,DO)
Shape(OBC,CB)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠COA=160°, O is the center of circle O. Find the measure of ∠CBA.
问题：如图所示，∠COA=160°，圆O的圆心为O。求∠CBA的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AO,OC,CB)
Shape(BC,OCB)
Shape(OA,OAC,CO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),160)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠AOB=110°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=110°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(OBA,AM,MB)
Shape(MA,AO,OM)
Shape(BM,MC,CB)
Shape(BC,OCB)
Shape(CM,MO,OA,OAC)
Collinear(AMC)
Collinear(BMO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠DAB=70°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ACD.
问题：如图所示，∠DAB=70°，O是圆O的圆心，AB是⊙O的直径。求∠ACD的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CM,MA)
Shape(AM,MD,DA)
Shape(AD,ODA)
Shape(DM,MO,OD)
Shape(DO,OB,BD)
Shape(MC,OCB,BO,OM)
Shape(DB,OBD)
Collinear(AMOB)
Collinear(CMD)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 20
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠BOC=110°, the center of circle O is O. Find the measure of ∠BAC.
问题：如图所示，∠BOC=110°，O是圆O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AC,CO,OB)
Shape(CA,OAC)
Shape(BO,OC,OCB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠BAO=40°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠BAO=40°，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(OA,OAC,CM,MO)
Shape(AO,OM,MA)
Shape(AM,MB,BA)
Shape(MC,CB,BM)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
sine_theorem(1,OBA)
triangle_property_angle_sum(1,OBA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠COB=110°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=110°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OM,MA,OAB,BO)
Shape(AM,MC,CA)
Shape(OB,BM,MO)
Shape(MB,BC,CM)
Shape(AC,OCA)
Shape(CB,OBC)
Collinear(AMB)
Collinear(OMC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),110)
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)
Problem: As shown in the diagram, ∠CAB=50°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=50°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠COA=80°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠CBA.
问题：如图所示，∠COA=80°，⊙O的圆心为O，圆O的直径为AB。求∠CBA的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(BO,OC,CB)
Shape(BC,OCB)
Shape(CO,OA,OAC)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),80)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠ABO=30°, ∠CAB=50°, the center of circle O is O. Find the measure of ∠ADC.
问题：如图所示，∠ABO=30°，∠CAB=50°，⊙O的圆心为O。求∠ADC的大小。
Construction (CDL):
Shape(OAB,BD,DA)
Shape(AD,DO,OC,CA)
Shape(OD,DB,BO)
Shape(OB,OBC,CO)
Shape(AC,OCA)
Collinear(ADB)
Collinear(DOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),30)
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
adjacent_complementary_angle(1,ADO,ODB)
adjacent_complementary_angle(1,COB,BOD)
triangle_property_angle_sum(1,ODB)
Problem: As shown in the diagram, ∠CAB=30°, ∠DPA=60°, O is the center of circle O. Find the measure of ∠DBA.
问题：如图所示，∠CAB=30°，∠DPA=60°，O是圆O的圆心。求∠DBA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AP,PC)
Shape(PA,OAD,DP)
Shape(OBC,CP,PB)
Shape(BP,PD,DB)
Shape(O)
Shape(BD,ODB)
Collinear(CPD)
Collinear(APB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
Equal(MeasureOfAngle(DPA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
adjacent_complementary_angle(1,DPA,APC)
triangle_property_angle_sum(1,APC)
Problem: As shown in the diagram, ∠COA=80°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BDC.
问题：如图所示，∠COA=80°，⊙O的圆心为O，AB是⊙O的直径。求∠BDC的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OC,CM,MO)
Shape(MB,BD,DM)
Shape(MC,OCB,BM)
Shape(DB,OBD)
Shape(AO,OM,MD,ODA)
Collinear(AOMB)
Collinear(CMD)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),80)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,MOC,COA)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠COA=50°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠CDA.
问题：如图所示，∠COA=50°，O是⊙O的圆心，AB是⊙O的直径。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AM,MD)
Shape(DM,MO,OB,OBD)
Shape(MA,OAC,CM)
Shape(MC,CO,OM)
Shape(OC,OCB,BO)
Collinear(AMOB)
Collinear(DMC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, the measure of ⌒OBC is 50, O is the center of circle O, AB is the diameter of circle O, AE⊥CE. Find the measure of ∠DGA.
问题：如图所示，弧OBC的角度为50，O是圆O的圆心，AB是圆O的直径，AE垂直于CE。求∠DGA的大小。
Construction (CDL):
Shape(OGA,AG)
Shape(GA,AO,OM,MG)
Shape(MO,OA,AD,DM)
Shape(OCG,GM,ME,EC)
Shape(EM,MD,DE)
Shape(CE,EB,OBC)
Shape(BE,ED,DB)
Shape(BD,ODB)
Shape(DA,OAD)
Collinear(AOMEB)
Collinear(GMD)
Collinear(CED)
Cocircular(O,ADBCG)
Conditions (CDL):
Equal(MeasureOfArc(OBC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(DGA))
Answer: 65
Theorem sequence:
vertical_angle(1,AEC,BED)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,BED)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,G)
Problem: As shown in the diagram, ∠BOA=50°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=50°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OB,OBC,CM,MO)
Shape(OM,MB,BO)
Shape(MA,OAB,BM)
Shape(MC,CA,AM)
Shape(AC,OCA)
Collinear(BMC)
Collinear(AMO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=90°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=90°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,OAB,BO)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 45
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ABO=30°, ∠CDA=20°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠ABO=30°，∠CDA=20°，O是圆O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(OA,AC,CO)
Shape(DO,OB,OBD)
Shape(OC,CB,BO)
Shape(BC,CA,OAB)
Collinear(DOC)
Collinear(ACB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),30)
Equal(MeasureOfAngle(CDA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
sine_theorem(1,ODA)
sine_theorem(1,OAB)
angle_addition(1,DAO,OAB)
Problem: As shown in the diagram, ∠AOB=50°, O is the center of ⊙O, BN is perpendicular to ON, DE⊥CE. Find the measure of ∠ECD.
问题：如图所示，∠AOB=50°，圆O的圆心为O，BN⊥ON，DE⊥CE。求∠ECD的大小。
Construction (CDL):
Shape(OAC,CM,MA)
Shape(AM,MN,NA)
Shape(OBA,AN,NB)
Shape(MC,CE,EM)
Shape(CP,PE,EC)
Shape(NM,ME,EP,PO,ON)
Shape(PC,CD,DP)
Shape(NO,OB,BN)
Shape(BO,OP,PD,ODB)
Shape(OCD,DC)
Collinear(AMEPD)
Collinear(ANO)
Collinear(CMNB)
Collinear(CPO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BN,ON)
PerpendicularBetweenLine(DE,CE)
Goal: Value(MeasureOfAngle(ECD))
Answer: 65
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)
altitude_of_triangle_judgment(1,ON,OBC)
isosceles_triangle_property_line_coincidence(1,OBC,N)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
triangle_property_angle_sum(1,ECD)
Problem: As shown in the diagram, ∠COB=70°, O is the center of ⊙O, the diameter of ⊙O is AB, DM⊥OM. Find the measure of ∠BAD.
问题：如图所示，∠COB=70°，O是⊙O的圆心，AB是圆O的直径，DM垂直于OM。求∠BAD的大小。
Construction (CDL):
Shape(OBC,CM,MB)
Shape(BM,MD,ODB)
Shape(MC,CO,OM)
Shape(MO,OD,DM)
Shape(DO,OA,AD)
Shape(DA,OAD)
Shape(OC,OCA,AO)
Collinear(AOMB)
Collinear(CMD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(DM,OM)
Goal: Value(MeasureOfAngle(BAD))
Answer: 35
Theorem sequence:
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)
altitude_of_triangle_judgment(1,OM,ODC)
isosceles_triangle_property_line_coincidence(1,ODC,M)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,A)
Problem: As shown in the diagram, ∠CAB=50°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=50°，O是圆O的圆心，AB是⊙O的直径。求∠ADC的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(AD,DM,MO,OA)
Shape(AO,OM,MC,CA)
Shape(OCA,AC)
Shape(MD,ODB,BM)
Shape(CM,MB,BC)
Shape(CB,OBC)
Collinear(AOMB)
Collinear(DMC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠ABD=65°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=65°，圆O的圆心为O，AB是圆O的直径。求∠DCB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OM,MD)
Shape(DM,MB,BD)
Shape(DB,OBD)
Shape(MO,OA,OAC,CM)
Shape(MC,CB,BM)
Shape(BC,OCB)
Collinear(AOMB)
Collinear(DMC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),65)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 25
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE)), DE=6, the radius of circle A is 5, A is the center of circle A. Find the length of line BC.
问题：如图所示，Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE))，DE=6，圆A的半径为5，A是⊙A的圆心。求直线BC的长度。
Construction (CDL):
Shape(ACB,BC)
Shape(AC,CB,BA)
Shape(ADC,CA,AD)
Shape(AB,ABE,EA)
Shape(DA,AE,ED)
Shape(DE,AED)
Cocircular(A,CBED)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BAC),MeasureOfAngle(DAE)),180)
Equal(LengthOfLine(DE),6)
Equal(RadiusOfCircle(A),5)
IsCentreOfCircle(A,A)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
radius_of_circle_property_length_equal(1,AC,A)
radius_of_circle_property_length_equal(1,AB,A)
radius_of_circle_property_length_equal(1,AE,A)
radius_of_circle_property_length_equal(1,AD,A)
cosine_theorem(1,AED)
cosine_theorem(1,ACB)
Problem: As shown in the diagram, ∠BOA=100°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=100°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OA,AB,BO)
Shape(BA,AC,CB)
Shape(BC,OCB)
Shape(CA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 130
Theorem sequence:
round_arc(1,OAB,OBA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BAC=25°, the center of ⊙O is O. Find the measure of ∠OCB.
问题：如图所示，∠BAC=25°，圆O的圆心为O。求∠OCB的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AM,MB)
Shape(BM,MC,CB)
Shape(BC,OCB)
Shape(CM,MO,OC)
Shape(OM,MA,OAC,CO)
Collinear(BMO)
Collinear(AMC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCB))
Answer: 65
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
triangle_property_angle_sum(1,OCB)
sine_theorem(1,OCB)
Problem: As shown in the diagram, ∠ACB=90°, ∠ACE=60°, ∠BAC=30°, O is the center of circle O. Find the measure of ∠AOE.
问题：如图所示，∠ACB=90°，∠ACE=60°，∠BAC=30°，圆O的圆心为O。求∠AOE的大小。
Construction (CDL):
Shape(OBE,EM,MB)
Shape(EO,OM,ME)
Shape(OE,OEA,AO)
Shape(BM,MC,CB)
Shape(MO,OA,AC,CM)
Shape(OAC,CA)
Shape(OCB,BC)
Collinear(EMC)
Collinear(AOMB)
Cocircular(O,BEAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(ACE),60)
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OEA,O)
arc_property_circumference_angle_external(1,OEA,C)
Problem: As shown in the diagram, ∠ABO=40°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=40°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OM,MC,OCA,AO)
Shape(OA,AM,MO)
Shape(CM,MB,BC)
Shape(MA,AB,BM)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
sine_theorem(1,OAB)
triangle_property_angle_sum(1,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ABD=38°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DCB.
问题：如图所示，∠ABD=38°，O是⊙O的圆心，AB是⊙O的直径。求∠DCB的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(DO,OB,BD)
Shape(DB,OBD)
Shape(BO,OC,CB)
Shape(BC,OCB)
Shape(OA,OAC,CO)
Collinear(AOB)
Collinear(DOC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),38)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 52
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
sine_theorem(1,OBD)
diameter_of_circle_judgment_pass_centre(1,COD,O)
diameter_of_circle_property_right_angle(1,CBD,O)
triangle_property_angle_sum(1,CBD)
Problem: As shown in the diagram, ∠COA=130°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BDC.
问题：如图所示，∠COA=130°，O是圆O的圆心，AB是圆O的直径。求∠BDC的大小。
Construction (CDL):
Shape(ODA,AO,OM,MD)
Shape(DM,MB,BD)
Shape(DB,OBD)
Shape(MO,OC,CM)
Shape(BM,MC,OCB)
Shape(OA,OAC,CO)
Collinear(AOMB)
Collinear(DMC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),130)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 25
Theorem sequence:
adjacent_complementary_angle(1,MOC,COA)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠CBA=55°, O is the center of circle O. Find the measure of ∠COA.
问题：如图所示，∠CBA=55°，O是圆O的圆心。求∠COA的大小。
Construction (CDL):
Shape(OM,MB,OBA,AO)
Shape(OA,AM,MO)
Shape(BM,MC,CB)
Shape(BC,OCB)
Shape(CM,MA,OAC)
Collinear(OMC)
Collinear(AMB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠AOC=140°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠CDB.
问题：如图所示，∠AOC=140°，圆O的圆心为O，圆O的直径为AB。求∠CDB的大小。
Construction (CDL):
Shape(ODB,BD)
Shape(DB,BM,MD)
Shape(DM,MO,OA,OAD)
Shape(MB,OBC,CM)
Shape(OM,MC,CO)
Shape(AO,OC,OCA)
Collinear(DMC)
Collinear(AOMB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),140)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,AOC,COM)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)