SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BOA=50°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=50°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CB)
Shape(OAB,BC,CA)
Shape(OAB,BO,OA)
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=60°, the center of ⊙O is O, the diameter of circle O is BD, AE is perpendicular to OE. Find the measure of ∠CDB.
问题：如图所示，∠BOA=60°，O是圆O的圆心，圆O的直径为BD，AE⊥OE。求∠CDB的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(DO,OE,EC,CD)
Shape(OAB,BE,EA)
Shape(OBC,CE,EB)
Shape(OCD,DC)
Shape(ODA,AO,OD)
Collinear(AEC)
Collinear(DOEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠ABC=40°, O is the center of circle O, CD is the diameter of ⊙O. Find the measure of ∠DCA.
问题：如图所示，∠ABC=40°，O是⊙O的圆心，圆O的直径为CD。求∠DCA的大小。
Construction (CDL):
Shape(CA,AD,DO,OC)
Shape(CA,AB,BC)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(ODB,BA,AD)
Shape(OAD,DA)
Collinear(COD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAD,O)
triangle_property_angle_sum(1,CAD)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠DOB=50°, the center of ⊙O is O, BE⊥OE. Find the measure of ∠CAD.
问题：如图所示，∠DOB=50°，O是圆O的圆心，BE⊥OE。求∠CAD的大小。
Construction (CDL):
Shape(OAB,OBD,DA)
Shape(OCA,AC)
Shape(OB,BE,EO)
Shape(OE,EC,CO)
Shape(OBD,DE,EB)
Shape(ODC,CA,AD)
Shape(ODC,CE,ED)
Collinear(BEC)
Collinear(OED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(MeasureOfAngle(CAD))
Answer: 25
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OBC,OED)
congruent_arc_judgment_length_equal(1,OBD,ODC)
congruent_arc_property_measure_equal(1,OBD,ODC)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,ODC,A)
Problem: As shown in the diagram, ∠ACO=45°, ∠OBA=30°, the center of circle O is O. Find the measure of ∠BOC.
问题：如图所示，∠ACO=45°，∠OBA=30°，圆O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(AC,CO,OB,BA)
Shape(OCB,BO,OC)
Shape(OAC,CA)
Shape(OBA,AB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACO),45)
Equal(MeasureOfAngle(OBA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 150
Theorem sequence:
round_angle(1,BOC,COB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
quadrilateral_property_angle_sum(1,ACOB)
Problem: As shown in the diagram, ∠CBO=70°, O is the center of circle O. Find the measure of ∠BAC.
问题：如图所示，∠CBO=70°，圆O的圆心为O。求∠BAC的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(BO,OC,CB)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OBA,AB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 20
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,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,BOC)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
Problem: As shown in the diagram, AC=AB, ∠AOC=60°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，AC=AB，∠AOC=60°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AO,OC,CA)
Shape(OCA,AC)
Shape(OAB,BA)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),LengthOfLine(AB))
Equal(MeasureOfAngle(AOC),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
Problem: As shown in the diagram, ∠BOA=80°, O is the center of ⊙O. Find the measure of ∠OCA.
问题：如图所示，∠BOA=80°，O是⊙O的圆心。求∠OCA的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OCA,AC)
Shape(OBC,CO,OB)
Shape(AO,OC,CA)
Collinear(BOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠BAC=45°, ∠OBA=60°, O is the center of circle O. Find the measure of ∠OCA.
问题：如图所示，∠BAC=45°，∠OBA=60°，⊙O的圆心为O。求∠OCA的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OC,CD,DO)
Shape(AO,OD,DA)
Shape(AD,DB,BA)
Shape(OCB,BD,DC)
Collinear(CDA)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),45)
Equal(MeasureOfAngle(OBA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 15
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
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)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,AOB)
triangle_property_angle_sum(1,AOC)
angle_addition(1,AOB,BOC)
Problem: As shown in the diagram, ∠BCA=60°, BD is the angle bisector of ∠ABC, O is the center of ⊙O, AC is the diameter of ⊙O. Find the measure of ∠DAB.
问题：如图所示，∠BCA=60°，BD平分∠ABC，O是圆O的圆心，AC是圆O的直径。求∠DAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AB,BD,DA)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DB,BC)
Shape(ODA,AD)
Collinear(AOC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),60)
IsBisectorOfAngle(BD,ABC)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 75
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
angle_addition(1,ABD,DBC)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, ∠ADC=36°, the center of circle O is O, BE is perpendicular to AE. Find the measure of ∠DAB.
问题：如图所示，∠ADC=36°，圆O的圆心为O，BE垂直于AE。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OE,EA)
Shape(CA,AE,EC)
Shape(CE,ED,DC)
Shape(OCA,AC)
Shape(ODC,CD)
Shape(OBD,DE,EO,OB)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),36)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,AE)
Goal: Value(MeasureOfAngle(DAB))
Answer: 54
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠AOB=50°, O is the center of circle O, BE is perpendicular to OE. Find the measure of ∠CDA.
问题：如图所示，∠AOB=50°，O是圆O的圆心，BE⊥OE。求∠CDA的大小。
Construction (CDL):
Shape(OBA,AE,EO,OB)
Shape(BE,EO,OB)
Shape(OAC,CD,DA)
Shape(OBA,OAC,CE,EB)
Shape(OCD,DC)
Shape(ODB,BO,OE,EA,AD)
Collinear(CEB)
Collinear(AEO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
Theorem sequence:
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)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠CAB=70°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠BCO.
问题：如图所示，∠CAB=70°，O是圆O的圆心，⊙O的直径为AB。求∠BCO的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CB)
Shape(CO,OB,BC)
Shape(CA,AO,OC)
Shape(OAB,BO,OA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 20
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
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, ⌒OAB is congruent to ⌒OEA, ∠BEA=80°, the center of circle O is O. Find the measure of ∠EAB.
问题：如图所示，弧OAB全等于弧OEA，∠BEA=80°，圆O的圆心为O。求∠EAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBE,EB)
Shape(OEA,AE)
Shape(AB,BE,EA)
Cocircular(O,ABE)
Conditions (CDL):
CongruentBetweenArc(OAB,OEA)
Equal(MeasureOfAngle(BEA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EAB))
Answer: 20
Theorem sequence:
congruent_arc_property_chord_equal(1,OAB,OEA)
isosceles_triangle_judgment_line_equal(1,ABE)
isosceles_triangle_property_angle_equal(1,ABE)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠CAB=32°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠ABC.
问题：如图所示，∠CAB=32°，O是⊙O的圆心，圆O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(AO,OB,BC,CA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OABB,BO,OA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 58
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠DCB=25°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠ABD.
问题：如图所示，∠DCB=25°，O是圆O的圆心，AB是圆O的直径。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(OE,ED,DO)
Shape(DE,EB,BD)
Shape(EC,CB,BE)
Shape(OAC,CE,EO,OA)
Shape(OCB,BC)
Shape(OBD,DB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,AOD,DOE)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_center_angle(1,ODA,O)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠BAC=60°, the radius of circle O is 1, O is the center of ⊙O, the tangent to circle O is PB, PC is the tangent to circle O. Find the length of line PC.
问题：如图所示，∠BAC=60°，⊙O的半径为1，O是⊙O的圆心，PB是⊙O的切线，PC是圆O的切线。求直线PC的长度。
Construction (CDL):
Shape(ODB,BO,OD)
Shape(DP,PB,ODB)
Shape(CP,PD,OCD)
Shape(OCD,DO,OC)
Shape(BA,AC,CO,OB)
Shape(OBA,AB)
Shape(OAC,CA)
Collinear(ODP)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),60)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PB,O)
IsTangentOfCircle(PC,O)
Goal: Value(LengthOfLine(PC))
Answer: sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PB,O,O)
tangent_of_circle_property_perpendicular(1,PC,O,O)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
mirror_congruent_triangle_judgment_hl(1,PBO,POC)
mirror_congruent_triangle_property_angle_equal(1,OPB,OCP)
angle_addition(1,BOP,POC)
triangle_property_angle_sum(1,CPO)
sine_theorem(1,CPO)
Problem: As shown in the diagram, AB=6, ∠CBA=30°, O is the center of circle O, the diameter of circle O is AB. Find the length of line BC.
问题：如图所示，AB=6，∠CBA=30°，⊙O的圆心为O，圆O的直径为AB。求直线BC的长度。
Construction (CDL):
Shape(CB,BO,OA,AC)
Shape(OCB,BC)
Shape(OAC,CA)
Shape(OBA,AO,OB)
Collinear(BOA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(BC))
Answer: 3*sqrt(3)
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,CBA)
sine_theorem(1,BAC)
Problem: As shown in the diagram, ∠CDE=50°, O is the center of circle O, the diameter of ⊙O is CD, AO∥DE. Find the measure of ∠CAO.
问题：如图所示，∠CDE=50°，O是⊙O的圆心，圆O的直径为CD，AO平行于DE。求∠CAO的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(AO,OC,CA)
Shape(OAD,DO,OA)
Shape(CO,OD,DE,OEC)
Shape(ODE,ED)
Collinear(COD)
Cocircular(O,ADEC)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(CAO))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
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, ∠COA=70°, the center of ⊙O is O, AB is the diameter of ⊙O, AD is parallel to CO. Find the measure of ∠AOD.
问题：如图所示，∠COA=70°，O是⊙O的圆心，圆O的直径为AB，AD平行于CO。求∠AOD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OD,DA,AO)
Shape(OAC,CO,OA)
Shape(OCB,BO,OC)
Shape(OBD,DO,OB)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(AOD))
Answer: 40
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
parallel_property_alternate_interior_angle(1,AD,CO)
triangle_property_angle_sum(1,ODA)
Problem: As shown in the diagram, ∠BOD=102°, O is the center of ⊙O, CD is the diameter of circle O. Find the measure of ∠CAB.
问题：如图所示，∠BOD=102°，⊙O的圆心为O，⊙O的直径为CD。求∠CAB的大小。
Construction (CDL):
Shape(ODB,BO,OD)
Shape(OBC,CO,OB)
Shape(OBC,CA,AB)
Shape(OCA,AC)
Shape(OCA,OAD,DO,OC)
Collinear(DOC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),102)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 39
Theorem sequence:
adjacent_complementary_angle(1,COB,BOD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠COB=120°, O is the center of ⊙O. Find the measure of ∠BAC.
问题：如图所示，∠COB=120°，O是⊙O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(OB,BC,CO)
Shape(BA,AC,CB)
Shape(OBA,AB)
Shape(OAC,CA)
Shape(OCB,BO,OC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_internal(1,OBC,A)
Problem: As shown in the diagram, ∠ABO=52°, ∠ACO=18°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠ABO=52°，∠ACO=18°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OBC,CD,DB)
Shape(OAB,BA)
Shape(OCA,AD,DO,OC)
Shape(AB,BD,DA)
Shape(DC,CO,OD)
Collinear(ADC)
Collinear(BDO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),52)
Equal(MeasureOfAngle(ACO),18)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 34
Theorem sequence:
arc_property_center_angle(1,OBC,O)
vertical_angle(1,BDA,ODC)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,DCO)
Problem: As shown in the diagram, the diameter of circle O is 8, ∠DBC=30°, O is the center of ⊙O, the diameter of circle O is AB, the diameter of ⊙O is DE. Find the length of line CD.
问题：如图所示，⊙O的直径为8，∠DBC=30°，O是⊙O的圆心，AB是⊙O的直径，⊙O的直径为DE。求直线CD的长度。
Construction (CDL):
Shape(ODA,AO,OB,BD)
Shape(CD,DB,BC)
Shape(DO,OE,EC,CD)
Shape(OEB,OBC,CE)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(OAE,OEB,BO,OA)
Collinear(AOB)
Collinear(EOD)
Cocircular(O,AEBCD)
Conditions (CDL):
Equal(DiameterOfCircle(O),8)
Equal(MeasureOfAngle(DBC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(DE,O)
Goal: Value(LengthOfLine(CD))
Answer: 4
Theorem sequence:
diameter_of_circle_property_length_equal(1,DE,O)
diameter_of_circle_property_right_angle(1,ECD,O)
arc_property_circumference_angle_external(1,OCD,E)
arc_property_circumference_angle_external(1,OCD,B)
sine_theorem(1,DEC)
Problem: As shown in the diagram, AB=BD, ∠ABD=70°, O is the center of ⊙O. Find the measure of ∠COB.
问题：如图所示，AB=BD，∠ABD=70°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OBC,CD,DB)
Shape(AB,BD,DA)
Shape(DC,CO,OD)
Shape(OAB,BA)
Shape(OCA,AD,DO,OC)
Collinear(ADC)
Collinear(BDO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(BD))
Equal(MeasureOfAngle(ABD),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 110
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BDA)
isosceles_triangle_property_angle_equal(1,BDA)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠CAB=60°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=60°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OAB,BA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ABD=50°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=50°，⊙O的圆心为O，AB是圆O的直径。求∠DCB的大小。
Construction (CDL):
Shape(OAC,OCB,BO,OA)
Shape(AO,OB,BD,DA)
Shape(DC,CB,BD)
Shape(OBD,DB)
Shape(OCB,BC)
Shape(ODA,AD)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, BC=6, O is the center of ⊙O, AB is the diameter of ⊙O, CD⊥OD. Find the length of line OD.
问题：如图所示，BC=6，O是圆O的圆心，AB是⊙O的直径，CD垂直于OD。求直线OD的长度。
Construction (CDL):
Shape(OCA,AD,DC)
Shape(DA,AO,OD)
Shape(CD,DO,OB,BC)
Shape(OBC,CB)
Shape(OAB,BO,OA)
Collinear(AOB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 3
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OD,CA)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
midsegment_of_triangle_judgment_midpoint(1,OD,ABC)
midsegment_of_triangle_property_length(1,OD,ABC)
Problem: As shown in the diagram, ∠COA=50°, the center of ⊙O is O, AE is perpendicular to OE. Find the measure of ∠BDC.
问题：如图所示，∠COA=50°，O是圆O的圆心，AE垂直于OE。求∠BDC的大小。
Construction (CDL):
Shape(OA,AE,EO)
Shape(OAC,CE,EA)
Shape(OCB,BE,EC)
Shape(OCB,BD,DC)
Shape(OBD,DB)
Shape(ODA,AO,OE,EC,CD)
Collinear(AEB)
Collinear(OEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(BDC))
Answer: 25
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OAB,OEC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠COB=100°, the center of ⊙O is O, AD is the diameter of ⊙O. Find the sum of the measure of ∠ABO and the measure of ∠OCA.
问题：如图所示，∠COB=100°，⊙O的圆心为O，AD是圆O的直径。求∠ABO的大小与∠OCA的大小之和。
Construction (CDL):
Shape(OAB,BA)
Shape(OCA,AC)
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Shape(OBD,DO,OB)
Shape(ODC,CO,OD)
Collinear(AOD)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(Add(MeasureOfAngle(ABO),MeasureOfAngle(OCA)))
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,OC,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
angle_addition(1,CAD,DAB)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, ∠BDC=35°, O is the center of ⊙O, AC is the diameter of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠BDC=35°，O是圆O的圆心，⊙O的直径为AC。求∠ACB的大小。
Construction (CDL):
Shape(ODC,CD)
Shape(ED,DC,CE)
Shape(EC,CB,BE)
Shape(AO,OE,EB,BA)
Shape(OBA,AB)
Shape(OCB,BC)
Shape(OAD,DE,EO,OA)
Collinear(AOEC)
Collinear(DEB)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
diameter_of_circle_property_right_angle(1,CBA,O)
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, the length of ⌒OAB is equal to the length of ⌒OBC, ∠BOA=70°, O is the center of ⊙O, BD is the diameter of ⊙O. Find the measure of ∠CDB.
问题：如图所示，⌒OAB的长度与⌒OBC的长度相等，∠BOA=70°，O是圆O的圆心，BD是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(OAB,BO,OA)
Shape(OBC,CD,DO,OB)
Shape(OCD,DC)
Collinear(DOB)
Cocircular(O,DABC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OBC))
Equal(MeasureOfAngle(BOA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 35
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠CAB=30°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BCO.
问题：如图所示，∠CAB=30°，O是圆O的圆心，AB是⊙O的直径。求∠BCO的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OC,CA,AO)
Shape(OB,BC,CO)
Shape(OBC,CB)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 60
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,OC,O)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OCA)
Problem: As shown in the diagram, ∠CAB=64°, the center of circle O is O, AC is the diameter of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CAB=64°，⊙O的圆心为O，圆O的直径为AC。求∠COB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(OBC,CO,OB)
Shape(CA,AB,BO,OC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),64)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 128
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠DAB=65°, O is the center of ⊙O, DE is perpendicular to OE. Find the measure of ∠BOC.
问题：如图所示，∠DAB=65°，⊙O的圆心为O，DE垂直于OE。求∠BOC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAC,CE,EA)
Shape(DA,AE,ED)
Shape(OBD,DE,EO,OB)
Shape(OE,EC,CO)
Shape(OC,OCB,BO)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),65)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,OE)
Goal: Value(MeasureOfAngle(BOC))
Answer: 130
Theorem sequence:
triangle_property_angle_sum(1,DAE)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
adjacent_complementary_angle(1,AED,DEO)
adjacent_complementary_angle(1,BOC,COA)