SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCA=40°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=40°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(OA,AE,EO)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, the diameter of circle O is 10, AB=8, CD=4, the center of ⊙O is O, CD is perpendicular to BD. Find the length of line BD.
问题：如图所示，圆O的直径为10，AB=8，CD=4，圆O的圆心为O，CD⊥BD。求直线BD的长度。
Construction (CDL):
Shape(HE,EC,OCH)
Shape(HA,AE,EH)
Shape(AH,OHA)
Shape(EA,AB,BE)
Shape(BA,OAB)
Shape(DE,EB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Collinear(HEOB)
Collinear(AEDC)
Cocircular(O,HABC)
Conditions (CDL):
Equal(DiameterOfCircle(O),10)
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(CD),4)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CD,BD)
Goal: Value(LengthOfLine(BD))
Answer: 16/3
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,HOB,O)
diameter_of_circle_property_right_angle(1,HAB,O)
diameter_of_circle_property_length_equal(1,HB,O)
right_triangle_judgment_angle(1,HAB)
right_triangle_property_pythagorean(1,HAB)
arc_property_circumference_angle_external(1,OAB,H)
arc_property_circumference_angle_external(1,OAB,C)
similar_triangle_judgment_aa(1,BCD,BHA)
similar_triangle_property_line_ratio(1,BCD,BHA)
similar_triangle_property_line_ratio(1,CDB,HAB)
Problem: As shown in the diagram, ∠BAC=55°, AB is the diameter of ⊙O. Find the measure of ∠CDA.
问题：如图所示，∠BAC=55°，AB是圆O的直径。求∠CDA的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(BC,OCB)
Shape(DO,OB,OBD)
Shape(AD,ODA)
Shape(AC,CE,EA)
Shape(AE,ED,DA)
Shape(EC,CB,BE)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),55)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, AB=8, BC=12, O is the center of circle O, the diameter of circle O is BC, ABCD is a rectangle. Find the length of line AE.
问题：如图所示，AB=8，BC=12，⊙O的圆心为O，BC是⊙O的直径，四边形ABCD是长方形。求直线AE的长度。
Construction (CDL):
Shape(CD,DA,AE,OCE)
Shape(EA,AB,OEB)
Shape(EO,OC,OCE)
Shape(BO,OE,OEB)
Shape(CB,OBC)
Collinear(BOC)
Collinear(AEO)
Cocircular(O,EBC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),12)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Rectangle(ABCD)
Goal: Value(LengthOfLine(AE))
Answer: 4
Theorem sequence:
radius_of_circle_property_length_equal(1,OE,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
right_triangle_judgment_angle(1,ABO)
right_triangle_property_pythagorean(1,ABO)
line_addition(1,BO,OC)
line_addition(1,AE,EO)
Problem: As shown in the diagram, ∠BCA=110°, O is the center of ⊙O. Find the measure of ∠AOB.
问题：如图所示，∠BCA=110°，O是圆O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CB,OBC)
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_internal(1,OBA,C)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CAB=40°，圆O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(CO,OB,OBC)
Shape(AC,OCA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠BCA=35°, O is the center of ⊙O. Find the measure of ∠ABO.
问题：如图所示，∠BCA=35°，O是⊙O的圆心。求∠ABO的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(OA,AE,EO)
Shape(EA,AB,BE)
Shape(CE,EB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
Problem: As shown in the diagram, ∠CAB=50°, the center of ⊙O is 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(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BC,CO)
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_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠CAB=58°, O is the center of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CAB=58°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,OBC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),58)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 116
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, AB=2, ∠ACB=30°, O is the center of circle O, the diameter of ⊙O is AD. Find the radius of circle O.
问题：如图所示，AB=2，∠ACB=30°，⊙O的圆心为O，圆O的直径为AD。求⊙O的半径。
Construction (CDL):
Shape(CA,OAC)
Shape(AB,OBA)
Shape(BD,ODB)
Shape(DE,EC,OCD)
Shape(BE,ED,DB)
Shape(AE,EB,BA)
Shape(AC,CE,EA)
Collinear(AEOD)
Collinear(CEB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfAngle(ACB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(RadiusOfCircle(O))
Answer: 2
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_circumference_angle_external(1,OBA,D)
diameter_of_circle_property_right_angle(1,DBA,O)
sine_theorem(1,ADB)
diameter_of_circle_property_length_equal(1,AD,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, AB=AC, ∠CAB=70°. Find the measure of ∠CDA.
问题：如图所示，AB=AC，∠CAB=70°。求∠CDA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),70)
Goal: Value(MeasureOfAngle(CDA))
Answer: 125
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠ABC=40°, O is the center of circle O. Find the measure of ∠CAO.
问题：如图所示，∠ABC=40°，O是⊙O的圆心。求∠CAO的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(CA,AO,OC)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
triangle_property_angle_sum(1,CAO)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, ∠BOA=80°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=80°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(AE,EO,OA)
Shape(BE,EA,OAB)
Shape(EB,BC,CE)
Shape(CB,OBC)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
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, ∠CAB=50°, the center of circle O is O. Find the measure of ∠BCO.
问题：如图所示，∠CAB=50°，O是⊙O的圆心。求∠BCO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(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 40
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, ∠CAB=30°, AB is the diameter of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠CAB=30°，圆O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠COB=80°, O is the center of circle 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(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),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠AOD=30°, O is the center of ⊙O. Find the measure of ∠BCD.
问题：如图所示，∠AOD=30°，O是⊙O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AO,OD,DA)
Shape(DO,OB,BC,CD)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 105
Theorem sequence:
adjacent_complementary_angle(1,AOD,DOB)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠BPA=45°, the radius of ⊙O is 2, O is the center of ⊙O. Find the length of line AB.
问题：如图所示，∠BPA=45°，⊙O的半径为2，圆O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(AP,OPA)
Shape(BA,OAB)
Shape(PB,OBP)
Shape(PA,AO,OB,BP)
Shape(OA,AB,BO)
Cocircular(O,ABP)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),45)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(2)
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,P)
arc_property_center_angle(1,OAB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_judgment_angle(1,BOA)
right_triangle_property_pythagorean(1,BOA)
Problem: As shown in the diagram, ∠COB=120°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=120°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AE,EO,OC,OCA)
Shape(AB,BE,EA)
Shape(OE,EC,CO)
Shape(EB,BC,CE)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠ABO=40°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=40°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(OA,AE,EO)
Shape(EA,AB,BE)
Shape(EB,BC,CE)
Collinear(AEC)
Collinear(OEB)
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)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠BCA=40°, O is the center of ⊙O, AC is the diameter of ⊙O. Find the measure of ∠BDC.
问题：如图所示，∠BCA=40°，⊙O的圆心为O，AC是⊙O的直径。求∠BDC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(DB,OBD)
Shape(CD,ODC)
Shape(AC,OCA)
Shape(AB,BC,CA)
Shape(BD,DC,CB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 130
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_internal(1,OBC,D)
Problem: As shown in the diagram, OA=6, ∠BCA=30°, the center of ⊙O is O, OD is the median of △ OAB. Find the length of line OD.
问题：如图所示，OA=6，∠BCA=30°，O是圆O的圆心，OD是三角形OAB的中线。求直线OD的长度。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(OA,AF,FO)
Shape(OF,FE,EO)
Shape(FA,AD,DF)
Shape(EF,FD,DB,BE)
Shape(CE,EB,BC)
Collinear(OFD)
Collinear(AFEC)
Collinear(OEB)
Collinear(ADB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(OA),6)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
IsMedianOfTriangle(OD,OAB)
Goal: Value(LengthOfLine(OD))
Answer: 3*sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_line_coincidence(2,OAB,D)
sine_theorem(1,OAD)
Problem: As shown in the diagram, ∠BOA=120°, O is the center of ⊙O. Find the measure of ∠APB.
问题：如图所示，∠BOA=120°，O是⊙O的圆心。求∠APB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,AP,PB,BO)
Shape(PA,OAP)
Shape(BP,OPB)
Cocircular(O,APB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(APB))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,P)
Problem: As shown in the diagram, ∠ADC=65°, P is the center of circle O, the diameter of circle O is AB. Find the measure of ∠CAB.
问题：如图所示，∠ADC=65°，圆O的圆心为P，AB是圆O的直径。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(CB,OBC)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(AD,DE,EA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),65)
IsCentreOfCircle(P,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 25
Theorem sequence:
arc_property_circumference_angle_external(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,CAB)
Problem: As shown in the diagram, AB=BC, AD=10, ∠ABC=120°, the diameter of ⊙O is AD. Find the length of line BD.
问题：如图所示，AB=BC，AD=10，∠ABC=120°，AD是⊙O的直径。求直线BD的长度。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DE,EC,OCD)
Shape(AE,ED,DA)
Shape(EA,AB,BE)
Shape(CE,EB,BC)
Collinear(DEB)
Collinear(AOD)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(BC))
Equal(LengthOfLine(AD),10)
Equal(MeasureOfAngle(ABC),120)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(BD))
Answer: 5*sqrt(3)
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
diameter_of_circle_property_right_angle(1,ABD,O)
triangle_property_angle_sum(1,ABD)
sine_theorem(1,DAB)
Problem: As shown in the diagram, Add(MeasureOfAngle(COB)=MeasureOfAngle(CAB)), the radius of ⊙O is 1, O is the center of circle O. Find the length of line BC.
问题：如图所示，Add(MeasureOfAngle(COB)=MeasureOfAngle(CAB))，⊙O的半径为1，圆O的圆心为O。求直线BC的长度。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(COB),MeasureOfAngle(CAB)),180)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
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)
sine_theorem(1,BCO)
Problem: As shown in the diagram, ∠CAB=40°, the center of ⊙O is O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=40°，O是⊙O的圆心。求∠OBC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(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, AB=4, BE=EC, ∠AOB=60°, O is the center of circle O. Find the length of line BC.
问题：如图所示，AB=4，BE=EC，∠AOB=60°，O是⊙O的圆心。求直线BC的长度。
Construction (CDL):
Shape(BO,OE,EC,OCB)
Shape(OB,BE,EO)
Shape(EB,BA,AE)
Shape(CE,EA,AC)
Shape(AB,OBA)
Shape(CA,OAC)
Collinear(OEA)
Collinear(BEC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BE),LengthOfLine(EC))
Equal(MeasureOfAngle(AOB),60)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: 4*sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,OBA)
circle_property_chord_perpendicular_bisect_chord(2,O,OE,BC)
adjacent_complementary_angle(1,AEB,BEO)
sine_theorem(1,BAE)
line_addition(1,BE,EC)
Problem: As shown in the diagram, AB=4, ∠BCA=30°, the center of circle O is O, BD is the diameter of ⊙O. Find the radius of ⊙O.
问题：如图所示，AB=4，∠BCA=30°，圆O的圆心为O，圆O的直径为BD。求圆O的半径。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DE,EC,OCD)
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Shape(CE,EB,BC)
Collinear(AEC)
Collinear(DEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(RadiusOfCircle(O))
Answer: 4
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
diameter_of_circle_property_right_angle(1,DAB,O)
sine_theorem(1,BDA)
diameter_of_circle_property_length_equal(1,BD,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, AB=BP, ∠BCA=30°, the radius of ⊙O is 5, the center of ⊙O is O. Find the length of line AP.
问题：如图所示，AB=BP，∠BCA=30°，圆O的半径为5，⊙O的圆心为O。求直线AP的长度。
Construction (CDL):
Shape(CA,AM,MC)
Shape(OM,ME,EN,NO)
Shape(MA,AE,EM)
Shape(EB,BN,NE)
Shape(EA,AB,BE)
Shape(PN,NB,BP)
Shape(OCA,AC)
Shape(AB,OBA)
Shape(BP,OPB)
Shape(OPC,CM,MO,ON,NP)
Collinear(CMEB)
Collinear(OMA)
Collinear(ONB)
Collinear(PNEA)
Cocircular(O,CABP)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(BP))
Equal(MeasureOfAngle(BCA),30)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AP))
Answer: 5*sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,P)
arc_property_center_angle(1,OAB,O)
cosine_theorem(1,OAB)
cosine_theorem(1,PAB)
Problem: As shown in the diagram, ∠CBA=70°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，∠CBA=70°，O是圆O的圆心。求∠COA的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AO,OC,CB)
Shape(OA,AC,CO)
Shape(BC,OCB)
Shape(CA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, the radius of circle O is 1, O is the center of circle O, AD⊥OD, CO is perpendicular to BO. Find the value of AE²+CE².
问题：如图所示，圆O的半径为1，O是圆O的圆心，AD⊥OD，CO垂直于BO。求AE²+CE²的值。
Construction (CDL):
Shape(AO,OE,EA)
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(EO,OM,ME)
Shape(OB,BM,MO)
Shape(EM,MC,CE)
Shape(MB,BC,CM)
Shape(OCA,AE,EC)
Shape(OAB,BD,DA)
Shape(CB,OBC)
Collinear(AEC)
Collinear(ADB)
Collinear(DOE)
Collinear(BME)
Collinear(OMC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
PerpendicularBetweenLine(CO,BO)
Goal: Value(Add(Pow(LengthOfLine(AE),2),Pow(LengthOfLine(EC),2)))
Answer: 2
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)
perpendicular_bisector_judgment_distance_equal(1,OD,AB)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
mirror_congruent_triangle_judgment_sas(1,DEA,DBE)
mirror_congruent_triangle_property_angle_equal(1,ADE,BED)
mirror_congruent_triangle_property_line_equal(1,DEA,DBE)
angle_addition(1,CAO,OAB)
angle_addition(1,ABO,OBE)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
vertical_angle(1,EMC,BMO)
triangle_property_angle_sum(1,MOB)
triangle_property_angle_sum(1,EMC)
right_triangle_judgment_angle(1,CEB)
right_triangle_property_pythagorean(1,CEB)
cosine_theorem(1,OBC)
Problem: As shown in the diagram, ∠ACB=40°, O is the center of ⊙O. Find the measure of ∠OBA.
问题：如图所示，∠ACB=40°，圆O的圆心为O。求∠OBA的大小。
Construction (CDL):
Shape(OAC,CM,MO,OA)
Shape(AO,OM,MA)
Shape(MC,CB,BM)
Shape(AM,MB,BA)
Shape(AB,OBA)
Shape(BC,OCB)
Collinear(OMB)
Collinear(AMC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBA))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
triangle_property_angle_sum(1,OBA)
sine_theorem(1,OBA)
Problem: As shown in the diagram, ∠CAB=30°, the radius of ⊙O is 2, the center of ⊙O is O. Find the length of line BC.
问题：如图所示，∠CAB=30°，⊙O的半径为2，O是⊙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(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: 2
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
cosine_theorem(1,OBC)
Problem: As shown in the diagram, ∠BCA=40°, the center of circle O is O, the diameter of circle O is AC. Find the measure of ∠CDB.
问题：如图所示，∠BCA=40°，O是⊙O的圆心，圆O的直径为AC。求∠CDB的大小。
Construction (CDL):
Shape(AB,BM,MO,OA)
Shape(MB,BC,CM)
Shape(DM,MC,CD)
Shape(OAB,BA)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(ODA,AO,OM,MD)
Collinear(AOMC)
Collinear(BMD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)