SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCA=50°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=50°，O是⊙O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, AB=8, AB=AC, the radius of ⊙O is 5, the center of ⊙O is O, the diameter of ⊙O is AD. Find the length of line EC.
问题：如图所示，AB=8，AB=AC，⊙O的半径为5，圆O的圆心为O，AD是圆O的直径。求直线EC的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BE,EO,OA)
Shape(AO,OE,EC,CA)
Shape(AC,OCA)
Shape(ED,DC,CE)
Shape(CD,ODC)
Shape(EB,OBD,DE)
Collinear(AOED)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(EC))
Answer: 24/5
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,AO,OD)
diameter_of_circle_property_right_angle(1,DCA,O)
right_triangle_judgment_angle(1,DCA)
right_triangle_property_pythagorean(1,DCA)
congruent_arc_judgment_chord_equal(1,OAB,OCA)
congruent_arc_property_measure_equal(1,OAB,OCA)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OAB,C)
mirror_similar_triangle_judgment_aa(1,ECA,CAD)
mirror_similar_triangle_property_line_ratio(1,ECA,CAD)
mirror_similar_triangle_property_line_ratio(1,AEC,ADC)
Problem: As shown in the diagram, ∠DAC=20°, △ABC is an equilateral △, O is the center of ⊙O. Find the measure of ∠ACD.
问题：如图所示，∠DAC=20°，△ABC为等边△，圆O的圆心为O。求∠ACD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BC,CA)
Shape(O)
Shape(CB,OBC)
Shape(AC,CD,DA)
Shape(AD,ODA)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),20)
EquilateralTriangle(ABC)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 40
Theorem sequence:
equilateral_triangle_property_angle(1,BCA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCD,A)
arc_addition_measure(1,OCD,ODA)
arc_property_circumference_angle_external(1,ODA,C)
Problem: As shown in the diagram, AC=8, AH=6, the radius of circle O is 5, O is the center of circle O, AH⊥CH. Find the length of line AB.
问题：如图所示，AC=8，AH=6，圆O的半径为5，圆O的圆心为O，AH⊥CH。求直线AB的长度。
Construction (CDL):
Shape(AB,BH,HA)
Shape(AH,HO,OA)
Shape(AO,OC,CA)
Shape(OH,HM,MO)
Shape(OM,MC,CO)
Shape(ME,EC,CM)
Shape(OAB,BA)
Shape(MH,HB,OBE,EM)
Shape(CE,OEC)
Shape(AC,OCA)
Collinear(AOME)
Collinear(BHMC)
Cocircular(O,ABEC)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(AH),6)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AH,CH)
Goal: Value(LengthOfLine(AB))
Answer: 15/2
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOE,O)
diameter_of_circle_property_right_angle(1,ECA,O)
arc_property_circumference_angle_external(1,OCA,E)
arc_property_circumference_angle_external(1,OCA,B)
adjacent_complementary_angle(1,BHA,AHC)
similar_triangle_judgment_aa(1,ABH,AEC)
circle_property_length_of_radius_and_diameter(1,O)
diameter_of_circle_property_length_equal(1,AE,O)
similar_triangle_property_line_ratio(1,BHA,ECA)
similar_triangle_property_line_ratio(1,HAB,CAE)
Problem: As shown in the diagram, ∠DAB=70°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠ACD.
问题：如图所示，∠DAB=70°，圆O的圆心为O，AB是圆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(MC,OCB,CO,OM)
Shape(MO,OB,BD,DM)
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, Add(MeasureOfAngle(CAB)=MeasureOfAngle(COB)), the radius of circle O is 2, O is the center of ⊙O. Find the length of line BC.
问题：如图所示，Add(MeasureOfAngle(CAB)=MeasureOfAngle(COB))，圆O的半径为2，O是圆O的圆心。求直线BC的长度。
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(Add(MeasureOfAngle(CAB),MeasureOfAngle(COB)),180)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: 2*sqrt(3)
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)
cosine_theorem(1,OBC)
Problem: As shown in the diagram, ∠CBA=60°, the center of ⊙O is O. Find the measure of ∠OAC.
问题：如图所示，∠CBA=60°，O是圆O的圆心。求∠OAC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CO,OA)
Shape(AO,OC,CB,BA)
Shape(BC,OCB)
Shape(AB,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
triangle_property_angle_sum(1,OAC)
sine_theorem(1,OAC)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of circle O, OD is perpendicular to CD. Find the measure of ∠DCO.
问题：如图所示，∠CAB=50°，⊙O的圆心为O，OD垂直于CD。求∠DCO的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(AC,OCA)
Shape(CD,DB,OBC)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,CD)
Goal: Value(MeasureOfAngle(DCO))
Answer: 40
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)
adjacent_complementary_angle(1,BDO,ODC)
perpendicular_bisector_judgment_distance_equal(1,OD,BC)
perpendicular_bisector_property_bisector(1,OD,BC)
angle_addition(1,COD,DOB)
triangle_property_angle_sum(1,ODC)
Problem: As shown in the diagram, ∠AOB=120°, the center of ⊙O is O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=120°，O是⊙O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(OA,OAC,CM,MO)
Shape(AO,OM,MA)
Shape(MC,CB,BM)
Shape(AM,MB,BA)
Shape(AB,OBA)
Shape(BC,OCB)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, AC=8, BM=4, the center of ⊙O is O, AE is perpendicular to BE, CD is perpendicular to AD. Find the radius of circle O.
问题：如图所示，AC=8，BM=4，圆O的圆心为O，AE垂直于BE，CD⊥AD。求圆O的半径。
Construction (CDL):
Shape(BN,NM,MB)
Shape(BM,MD,DB)
Shape(NA,AM,MN)
Shape(MA,AK,KM)
Shape(KA,AE,EK)
Shape(MK,KQ,QM)
Shape(DM,MQ,QO,OP,PD)
Shape(QK,KE,EC,CQ)
Shape(PO,OQ,QC,CP)
Shape(HB,BD,DP,PH)
Shape(HP,PC,CH)
Shape(OBA,AN,NB)
Shape(OAC,CE,EA)
Shape(HC,OCH)
Shape(BH,OHB)
Collinear(BNA)
Collinear(BMKE)
Collinear(BDPC)
Collinear(AMD)
Collinear(AKQOPH)
Collinear(AEC)
Collinear(CQMN)
Cocircular(O,ACHB)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(BM),4)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,BE)
PerpendicularBetweenLine(CD,AD)
Goal: Value(RadiusOfCircle(O))
Answer: 2*sqrt(5)
Theorem sequence:
altitude_of_triangle_judgment(1,BE,BAC)
altitude_of_triangle_judgment(1,AD,ACB)
orthocenter_of_triangle_judgment_intersection(1,M,CBA,D,E)
orthocenter_of_triangle_property_intersection(1,M,CBA,N)
diameter_of_circle_judgment_pass_centre(1,AOH,O)
diameter_of_circle_property_right_angle(1,HBA,O)
diameter_of_circle_property_right_angle(1,ACH,O)
parallel_judgment_ipsilateral_internal_angle(1,BH,NC)
parallel_judgment_ipsilateral_internal_angle(1,EB,CH)
parallel_property_collinear_extend(3,EB,CH,M)
parallel_property_collinear_extend(3,CN,HB,M)
parallelogram_judgment_parallel_and_parallel(1,BMCH)
parallelogram_property_opposite_line_equal(1,BMCH)
right_triangle_judgment_angle(1,ACH)
right_triangle_property_pythagorean(1,ACH)
diameter_of_circle_property_length_equal(1,AH,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, AC=6, OC=5, the center of ⊙O is O, the diameter of circle O is AB. Find the length of line BC.
问题：如图所示，AC=6，OC=5，圆O的圆心为O，AB是圆O的直径。求直线BC的长度。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(BO,OC,CB)
Shape(OA,AC,CO)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(OC),5)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(BC))
Answer: 8
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)
line_addition(1,BO,OA)
diameter_of_circle_property_right_angle(1,ACB,O)
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
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(OM,MC,OCA,AO)
Shape(OA,AM,MO)
Shape(MB,BC,CM)
Shape(MA,AB,BM)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(OMB)
Collinear(AMC)
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, BC=6, OA=2, ∠CAB=90°, the center of circle O is O, ∠CAB is the right angle of the isosceles right triangle ABC. Find the radius of circle O.
问题：如图所示，BC=6，OA=2，∠CAB=90°，O是⊙O的圆心，AB和AC是等腰直角三角形ABC的腰。求圆O的半径。
Construction (CDL):
Shape(AC,OCB,BA)
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Cocircular(O,BC)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(OA),2)
Equal(MeasureOfAngle(CAB),90)
IsCentreOfCircle(O,O)
IsoscelesRightTriangle(ABC)
Goal: Value(RadiusOfCircle(O))
Answer: sqrt(10)
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
mirror_congruent_triangle_judgment_sss(1,ABO,AOC)
mirror_congruent_triangle_property_angle_equal(1,ABO,AOC)
angle_addition(1,COA,OAB)
right_triangle_property_pythagorean(1,CAB)
cosine_theorem(1,ABO)
Problem: As shown in the diagram, AB=10, ∠OBA=90°, ∠XOY=45°. Find the length of line OA.
问题：如图所示，AB=10，∠OBA=90°，∠XOY=45°。求直线OA的长度。
Construction (CDL):
Shape(AO,OB,BA)
Shape(CA,AB,BC)
Shape(XA,AC)
Shape(CB,BY)
Collinear(OAX)
Collinear(OBY)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(MeasureOfAngle(OBA),90)
Equal(MeasureOfAngle(XOY),45)
Goal: Value(LengthOfLine(OA))
Answer: 10*sqrt(2)
Theorem sequence:
sine_theorem(1,AOB)
Problem: As shown in the diagram, ∠CAB=70°, O is the center of circle O. Find the measure of ∠COB.
问题：如图所示，∠CAB=70°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CAB=75°, P is the center of ⊙P. Find the measure of ∠CPB.
问题：如图所示，∠CAB=75°，P是⊙P的圆心。求∠CPB的大小。
Construction (CDL):
Shape(AB,BP,PC,CA)
Shape(PB,BC,CP)
Shape(PAB,BA)
Shape(PBC,CB)
Shape(AC,PCA)
Cocircular(P,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),75)
IsCentreOfCircle(P,P)
Goal: Value(MeasureOfAngle(CPB))
Answer: 150
Theorem sequence:
arc_property_center_angle(1,PBC,P)
arc_property_circumference_angle_external(1,PBC,A)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of circle O. Find the measure of ∠COB.
问题：如图所示，∠CAB=50°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(AC,OCA)
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, ∠OAB=28°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=28°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),28)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 62
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, ∠CAB=80°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=80°，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),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 160
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ABC=45°, ∠CAB=70°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠ABC=45°，∠CAB=70°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OB,BC,CA)
Shape(OAB,BA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(CAB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 130
Theorem sequence:
triangle_property_angle_sum(1,ABC)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, the radius of circle O is 2, triangle ABC is an equilateral triangle, O is the center of circle O. Find the length of line AB.
问题：如图所示，圆O的半径为2，三角形ABC为等边三角形，O是⊙O的圆心。求直线AB的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OB,BC,CA)
Shape(OAB,BA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(RadiusOfCircle(O),2)
EquilateralTriangle(ABC)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(3)
Theorem sequence:
equilateral_triangle_property_angle(1,CAB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
cosine_theorem(1,OAB)
Problem: As shown in the diagram, the radius of ⊙O is 2, Sin(MeasureOfAngle(ABC))=3/4, O is the center of circle O. Find the length of line AC.
问题：如图所示，圆O的半径为2，Sin(MeasureOfAngle(ABC))=3/4，圆O的圆心为O。求直线AC的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BC,CO,OA)
Shape(AO,OC,CA)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(RadiusOfCircle(O),2)
Equal(Sin(MeasureOfAngle(ABC)),3/4)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AC))
Answer: 3
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,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)
cosine_theorem(1,OCA)
Problem: As shown in the diagram, ∠COA=130°, the center of ⊙O is O. Find the measure of ∠DBA.
问题：如图所示，∠COA=130°，⊙O的圆心为O。求∠DBA的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(OA,AC,CO)
Shape(AB,BC,CA)
Shape(OAB,BA)
Shape(CB,OBC)
Shape(DB,BA)
Collinear(DBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 65
Theorem sequence:
arc_property_center_angle(1,OAC,O)
round_arc(1,OAC,OCA)
arc_property_circumference_angle_external(1,OCA,B)
adjacent_complementary_angle(1,DBA,ABC)
Problem: As shown in the diagram, ∠DAB=135°, O is the center of circle O. Find the measure of ∠ODB.
问题：如图所示，∠DAB=135°，O是⊙O的圆心。求∠ODB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(DA,AB,BD)
Shape(DB,BO,OD)
Shape(DO,OB,OBD)
Cocircular(O,DAB)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),135)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ODB))
Answer: 45
Theorem sequence:
arc_property_center_angle(1,ODB,O)
round_arc(1,OBD,ODB)
arc_property_circumference_angle_external(1,OBD,A)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OB,O)
triangle_property_angle_sum(1,BOD)
sine_theorem(1,ODB)
Problem: As shown in the diagram, ∠OBC=40°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=40°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OC,OCA,AD,DO)
Shape(CO,OD,DC)
Shape(DA,AB,BD)
Shape(DB,BC,CD)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
sine_theorem(1,OBC)
triangle_property_angle_sum(1,OBC)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOA=110°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=110°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(BA,OAB)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, AB=AC, ∠ABC=70°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，AB=AC，∠ABC=70°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(OAB,BA)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
sine_theorem(1,ABC)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, AB=12, OC=10, the center of ⊙O is O, OM⊥BM. Find the length of line MC.
问题：如图所示，AB=12，OC=10，圆O的圆心为O，OM⊥BM。求直线MC的长度。
Construction (CDL):
Shape(OB,OBA,AO)
Shape(OA,AM,MO)
Shape(OM,MB,BO)
Shape(MA,OAC,CM)
Shape(MC,OCB,BM)
Collinear(OMC)
Collinear(AMB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(OC),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OM,BM)
Goal: Value(LengthOfLine(MC))
Answer: 2
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)
adjacent_complementary_angle(1,AMO,OMB)
isosceles_triangle_judgment_line_equal(1,OAB)
altitude_of_triangle_judgment(1,OM,OAB)
isosceles_triangle_property_line_coincidence(1,OAB,M)
line_addition(1,AM,MB)
right_triangle_judgment_angle(1,AMO)
right_triangle_property_pythagorean(1,AMO)
line_addition(1,OM,MC)
Problem: As shown in the diagram, AB=13, AC=5, BC=12, O is the center of circle O, BM is the tangent to ⊙O. Find the length of line PQ.
问题：如图所示，AB=13，AC=5，BC=12，圆O的圆心为O，BM是圆O的切线。求直线PQ的长度。
Construction (CDL):
Shape(BQ,OMQ,MB)
Shape(OMQ,QO,OM)
Shape(QC,CO,OQ)
Shape(OQC,CQ)
Shape(OC,CP,PO)
Shape(PC,OCP)
Shape(MO,OP,OPM)
Shape(OPM,PA,AM)
Collinear(POQ)
Collinear(COM)
Collinear(BQC)
Collinear(CPA)
Collinear(AMB)
Cocircular(O,CPMQ)
Conditions (CDL):
Equal(LengthOfLine(AB),13)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(BC),12)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BM,O)
Goal: Value(LengthOfLine(PQ))
Answer: 60/13
Theorem sequence:
right_triangle_judgment_pythagorean_inverse(1,BCA)
tangent_of_circle_property_perpendicular(1,BM,O,O)
sine_theorem(1,ABC)
sine_theorem(1,CMB)
diameter_of_circle_judgment_pass_centre(1,COM,O)
diameter_of_circle_judgment_pass_centre(1,POQ,O)
diameter_of_circle_property_length_equal(1,CM,O)
diameter_of_circle_property_length_equal(1,PQ,O)
Problem: As shown in the diagram, the radius of circle O is 5, the center of circle O is O, AO⊥MO. Find the length of line OA.
问题：如图所示，圆O的半径为5，O是⊙O的圆心，AO垂直于MO。求直线OA的长度。
Construction (CDL):
Shape(ONM,MO,ON)
Shape(OM,OMA,AO)
Shape(OA,OAN,NO)
Collinear(MON)
Cocircular(O,MAN)
Conditions (CDL):
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AO,MO)
Goal: Value(LengthOfLine(OA))
Answer: 5
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
Problem: As shown in the diagram, OP=5, ∠BOA=30°, P is the center of circle P, OF is the tangent to circle P. Find the radius of ⊙P.
问题：如图所示，OP=5，∠BOA=30°，P是⊙P的圆心，OF是圆O的切线。求圆P的半径。
Construction (CDL):
Shape(FO,OE,PFE)
Shape(PFE,EP,PF)
Shape(FP,PA,PAF)
Shape(AP,PE,PEA)
Shape(BF,FP)
Collinear(OFB)
Collinear(OEPA)
Cocircular(P,FEA)
Conditions (CDL):
Equal(LengthOfLine(OP),5)
Equal(MeasureOfAngle(BOA),30)
IsCentreOfCircle(P,P)
IsTangentOfCircle(OF,P)
Goal: Value(RadiusOfCircle(P))
Answer: 5/2
Theorem sequence:
tangent_of_circle_property_perpendicular(1,OF,P,P)
sine_theorem(1,PFO)
radius_of_circle_property_length_equal(1,PF,P)
Problem: As shown in the diagram, ∠BAO=20°, the center of ⊙O is O, the tangent to circle O is CB, AO is perpendicular to CO. Find the measure of ∠BCO.
问题：如图所示，∠BAO=20°，圆O的圆心为O，CB是⊙O的切线，AO垂直于CO。求∠BCO的大小。
Construction (CDL):
Shape(OA,OAB,BO)
Shape(OB,BD,DO)
Shape(AO,OD,DA)
Shape(AD,DM,OMA)
Shape(DB,OBM,MD)
Shape(OBM,BC,CM)
Collinear(ODMC)
Collinear(ADB)
Cocircular(O,BMA)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),20)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CB,O)
PerpendicularBetweenLine(AO,CO)
Goal: Value(MeasureOfAngle(BCO))
Answer: -70+180*acos(-sin(pi/9))/pi
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)
angle_addition(1,AOC,COB)
tangent_of_circle_property_perpendicular(1,CB,O,O)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠ADC=32°, O is the center of ⊙O, BA is the tangent to circle O. Find the measure of ∠OBA.
问题：如图所示，∠ADC=32°，O是圆O的圆心，圆O的切线为BA。求∠OBA的大小。
Construction (CDL):
Shape(DM,MO,OA,OAD)
Shape(DC,CM,MD)
Shape(OM,MA,AO)
Shape(MC,OCA,AM)
Shape(CD,ODC)
Shape(OCA,CB,BA)
Collinear(DMA)
Collinear(OMCB)
Cocircular(O,DCA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),32)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(OBA))
Answer: 26
Theorem sequence:
tangent_of_circle_property_perpendicular(2,BA,O,O)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,AOB)
Problem: As shown in the diagram, ∠ADE=65°, O is the center of ⊙O, the tangent to circle O is FC. Find the measure of ∠AFC.
问题：如图所示，∠ADE=65°，圆O的圆心为O，圆O的切线为FC。求∠AFC的大小。
Construction (CDL):
Shape(OC,OCA,AO)
Shape(CO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(OBC,BF,FC)
Shape(EO,OA,AD,DE)
Shape(ED,ODB,BE)
Shape(DA,OAD)
Collinear(AOEBF)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),65)
IsCentreOfCircle(O,O)
IsTangentOfCircle(FC,O)
Goal: Value(MeasureOfAngle(AFC))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
tangent_of_circle_property_perpendicular(2,FC,O,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
sine_theorem(1,OBC)
triangle_property_angle_sum(1,COF)
triangle_property_angle_sum(1,COB)