SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, the radius of circle O is 3, the radius of circle Z is 5, the center of circle O is O, the center of circle Z is O, the tangent to ⊙O is AC, the tangent to circle O is BC. Find the length of line AB.
问题：如图所示，圆O的半径为3，⊙Z的半径为5，圆O的圆心为O，O是⊙Z的圆心，⊙O的切线为AC，⊙O的切线为BC。求直线AB的长度。
Construction (CDL):
Shape(ODC,CO,OD)
Shape(OCE,EO,OC)
Shape(DA,AC,ODC)
Shape(CB,BE,OCE)
Shape(OAB,BC,CA)
Shape(OBA,AC,CB)
Shape(OED,DO,OE)
Collinear(ADO)
Collinear(BEO)
Collinear(ACB)
Cocircular(O,DCE)
Cocircular(Z,AB)
Conditions (CDL):
Equal(RadiusOfCircle(O),3)
Equal(RadiusOfCircle(Z),5)
IsCentreOfCircle(O,O)
IsCentreOfCircle(O,Z)
IsTangentOfCircle(AC,O)
IsTangentOfCircle(BC,O)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,Z)
radius_of_circle_property_length_equal(1,OB,Z)
tangent_of_circle_property_perpendicular(2,AC,O,O)
tangent_of_circle_property_perpendicular(1,BC,O,O)
right_triangle_judgment_angle(1,ACO)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,ACO)
right_triangle_property_pythagorean(1,OCB)
line_addition(1,AC,CB)
Problem: As shown in the diagram, AB=4, OM=1, the center of ⊙O is O, PA is the tangent to ⊙O, BM⊥OM. Find the length of line PA.
问题：如图所示，AB=4，OM=1，⊙O的圆心为O，PA是⊙O的切线，BM垂直于OM。求直线PA的长度。
Construction (CDL):
Shape(OBC,CM,MB)
Shape(OM,MA,AO)
Shape(OCA,AM,MC)
Shape(OAB,BM,MO,OA)
Shape(CP,PA,OCA)
Collinear(OMCP)
Collinear(BMA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(OM),1)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
PerpendicularBetweenLine(BM,OM)
Goal: Value(LengthOfLine(PA))
Answer: 2*sqrt(5)
Theorem sequence:
line_addition(1,OM,MP)
line_addition(1,BM,MA)
circle_property_chord_perpendicular_bisect_chord(1,O,OM,BA)
tangent_of_circle_property_perpendicular(2,PA,O,O)
vertical_angle(1,BMO,AMC)
right_triangle_judgment_angle(1,OMA)
right_triangle_judgment_angle(1,AMP)
right_triangle_judgment_angle(1,PAO)
right_triangle_property_pythagorean(1,OMA)
right_triangle_property_pythagorean(1,AMP)
right_triangle_property_pythagorean(1,PAO)
Problem: As shown in the diagram, the diameter of circle O is 2, ∠DAO=30°, O is the center of circle O, the diameter of ⊙O is AB, the tangent to ⊙O is CD. Find the length of line CD.
问题：如图所示，圆O的直径为2，∠DAO=30°，⊙O的圆心为O，圆O的直径为AB，⊙O的切线为CD。求直线CD的长度。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBD,DO,OB)
Shape(AO,OD,DA)
Shape(ODA,AD)
Shape(BC,CD,OBD)
Collinear(AOBC)
Cocircular(O,DAB)
Conditions (CDL):
Equal(DiameterOfCircle(O),2)
Equal(MeasureOfAngle(DAO),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Goal: Value(LengthOfLine(CD))
Answer: sqrt(3)
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OD,O)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_center_angle(1,OBD,O)
tangent_of_circle_property_perpendicular(2,CD,O,O)
triangle_property_angle_sum(1,DOC)
sine_theorem(1,DOC)
Problem: As shown in the diagram, ∠CAE=120°, the center of circle O is O, AD is the diameter of circle O, the tangent to circle O is EA. Find the measure of ∠ABC.
问题：如图所示，∠CAE=120°，圆O的圆心为O，圆O的直径为AD，⊙O的切线为EA。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBD,DF,FB)
Shape(AB,BF,FO,OA)
Shape(ODC,CD)
Shape(OCA,AC)
Shape(FD,DC,CF)
Shape(AO,OF,FC,CA)
Shape(BA,AE)
Collinear(BFC)
Collinear(AOFD)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAE),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
IsTangentOfCircle(EA,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 60
Theorem sequence:
angle_addition(1,CAF,FAE)
tangent_of_circle_property_perpendicular(1,EA,O,O)
diameter_of_circle_property_right_angle(1,DCA,O)
triangle_property_angle_sum(1,DCA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠CEA=25°, the center of circle O is O, the diameter of ⊙O is AB, DA is the tangent to circle O. Find the measure of ∠ADC.
问题：如图所示，∠CEA=25°，O是⊙O的圆心，AB是⊙O的直径，DA是⊙O的切线。求∠ADC的大小。
Construction (CDL):
Shape(OBE,EF,FO,OB)
Shape(OEA,AE)
Shape(FE,EA,AF)
Shape(CB,BO,OF,FC)
Shape(OCB,BC)
Shape(OAC,CF,FA)
Shape(AD,DC,OAC)
Collinear(BOFA)
Collinear(EFC)
Collinear(BCD)
Cocircular(O,EACB)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 65
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DA,O,O)
arc_property_circumference_angle_external(1,OAC,E)
arc_property_circumference_angle_external(1,OAC,B)
triangle_property_angle_sum(1,BAD)
Problem: As shown in the diagram, CD=4, O is the center of ⊙O, AB is the diameter of ⊙O, CD is the tangent to circle O, quadrilateral OCDE is a parallelogram. Find the length of line ED.
问题：如图所示，CD=4，O是圆O的圆心，AB是⊙O的直径，CD是⊙O的切线，CO和DE是▱OCDE的一组对边。求直线ED的长度。
Construction (CDL):
Shape(OEA,AO,OE)
Shape(ODE,ED)
Shape(OBD,DO,OB)
Shape(OD,DE,EO)
Shape(BC,CD,OBD)
Collinear(AOBC)
Cocircular(O,BDEA)
Conditions (CDL):
Equal(LengthOfLine(CD),4)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Parallelogram(OCDE)
Goal: Value(LengthOfLine(ED))
Answer: 4*sqrt(2)
Theorem sequence:
parallelogram_property_opposite_line_equal(1,CDEO)
tangent_of_circle_property_perpendicular(2,CD,O,O)
parallel_property_alternate_interior_angle(1,OE,CD)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OE,O)
right_triangle_judgment_angle(1,EOD)
right_triangle_property_pythagorean(1,EOD)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of circle O, AD is the tangent to ⊙O, AF is the tangent to circle O, BE is the tangent to ⊙O. Find the measure of ∠DEF.
问题：如图所示，∠CAB=50°，O是圆O的圆心，AD是圆O的切线，⊙O的切线为AF，⊙O的切线为BE。求∠DEF的大小。
Construction (CDL):
Shape(AD,OFD,FA)
Shape(OFD,DO,OF)
Shape(DB,BE,ODE)
Shape(ODE,ED)
Shape(OD,DE,EF,FO)
Shape(OEF,FE)
Shape(OEF,EC,CF)
Collinear(ADB)
Collinear(BEC)
Collinear(AFC)
Cocircular(O,DEF)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AD,O)
IsTangentOfCircle(AF,O)
IsTangentOfCircle(BE,O)
Goal: Value(MeasureOfAngle(DEF))
Answer: 65
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AD,O,O)
tangent_of_circle_property_perpendicular(1,AF,O,O)
quadrilateral_property_angle_sum(1,FADO)
arc_property_center_angle(1,OFD,O)
arc_property_circumference_angle_external(1,OFD,E)
Problem: As shown in the diagram, OP=6, ∠BAC=50°, the radius of ⊙O is 3, O is the center of ⊙O, the tangent to ⊙O is PA, the tangent to ⊙O is PB. Find the length of line PB.
问题：如图所示，OP=6，∠BAC=50°，⊙O的半径为3，圆O的圆心为O，PA是⊙O的切线，圆O的切线为PB。求直线PB的长度。
Construction (CDL):
Shape(PA,AO,OP)
Shape(PB)
Shape(O)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OP),6)
Equal(MeasureOfAngle(BAC),50)
Equal(RadiusOfCircle(O),3)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(LengthOfLine(PB))
Answer: 3*sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
tangent_of_circle_property_perpendicular(2,PA,O,O)
right_triangle_judgment_angle(1,PAO)
right_triangle_property_pythagorean(1,PAO)
tangent_of_circle_property_length_equal(1,PA,PB,O)
Problem: As shown in the diagram, ∠APB=70°, the center of circle O is O, the tangent to ⊙O is PA, the tangent to circle O is PB. Find the measure of ∠ACB.
问题：如图所示，∠APB=70°，O是⊙O的圆心，⊙O的切线为PA，⊙O的切线为PB。求∠ACB的大小。
Construction (CDL):
Shape(AP,PB,OCB,OAC)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OA,AC,CB,BO)
Shape(OB,BD,DA,AO)
Shape(DB,OBD)
Shape(AD,ODA)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(APB),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 125
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
quadrilateral_property_angle_sum(1,APBO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, OA=6, PA=8, the center of ⊙O is O, PA is the tangent to ⊙O. Find the value of cos(AP).
问题：如图所示，OA=6，PA=8，O是圆O的圆心，圆O的切线为PA。求cos(AP)的值。
Construction (CDL):
Shape(AP,PO,OA)
Shape(O)
Cocircular(O,A)
Conditions (CDL):
Equal(LengthOfLine(OA),6)
Equal(LengthOfLine(PA),8)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(Cos(MeasureOfAngle(APO)))
Answer: 4/5
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
right_triangle_judgment_angle(1,OAP)
right_triangle_property_pythagorean(1,OAP)
cosine_theorem(1,POA)
Problem: As shown in the diagram, OA=OB, ∠ABO=20°, the center of ⊙O is O, BE is the tangent to circle O, CA is the tangent to circle O. Find the measure of ∠OCA.
问题：如图所示，OA=OB，∠ABO=20°，圆O的圆心为O，⊙O的切线为BE，⊙O的切线为CA。求∠OCA的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Shape(OAB,BA)
Collinear(BOC)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OA),LengthOfLine(OB))
Equal(MeasureOfAngle(ABO),20)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BE,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 50
Theorem sequence:
tangent_of_circle_property_perpendicular(2,CA,O,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,ABC)
angle_addition(1,CAO,OAB)
Problem: As shown in the diagram, ∠ABO=20°, the center of ⊙O is O, CB is the tangent to ⊙O. Find the measure of ∠BCO.
问题：如图所示，∠ABO=20°，O是⊙O的圆心，CB是圆O的切线。求∠BCO的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BD,DO)
Shape(DB,OBD)
Shape(OBD,BC,CD)
Shape(OAB,BO)
Shape(ODA,AO,OD)
Collinear(AODC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),20)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 50
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CB,O,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,ABC)
angle_addition(1,ABO,OBC)
Problem: As shown in the diagram, ∠BPA=40°, the center of ⊙O is O, PA is the tangent to ⊙O, the tangent to ⊙O is PB. Find the measure of ∠ACB.
问题：如图所示，∠BPA=40°，⊙O的圆心为O，PA是⊙O的切线，圆O的切线为PB。求∠ACB的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BC)
Shape(OC,CB,BO)
Shape(OB,BA,AO)
Shape(AB,OBA)
Shape(OBA,BP,PA)
Collinear(AOC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 70
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
quadrilateral_property_angle_sum(1,AOBP)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, OC=OB, ∠COA=80°, O is the center of circle O, DA is the tangent to circle O. Find the measure of ∠ADB.
问题：如图所示，OC=OB，∠COA=80°，⊙O的圆心为O，⊙O的切线为DA。求∠ADB的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(BO,OC,CB)
Shape(OCB,BC)
Shape(OA,OAC,CO)
Shape(OAC,AD,DC)
Collinear(BOA)
Collinear(BCD)
Cocircular(O,BAC)
Conditions (CDL):
Equal(LengthOfLine(OC),LengthOfLine(OB))
Equal(MeasureOfAngle(COA),80)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ADB))
Answer: 50
Theorem sequence:
flat_angle(1,BOA)
angle_addition(1,BOC,COA)
triangle_property_angle_sum(1,BOC)
triangle_property_angle_sum(1,BAD)
tangent_of_circle_property_perpendicular(1,DA,O,O)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
Problem: As shown in the diagram, OD=OB, ∠BAC=55°, O is the center of circle O, the tangent to circle O is AC. Find the measure of ∠COD.
问题：如图所示，OD=OB，∠BAC=55°，O是⊙O的圆心，AC是⊙O的切线。求∠COD的大小。
Construction (CDL):
Shape(OCB,BO,OC)
Shape(OBD,DB)
Shape(OB,BD,DO)
Shape(OD,ODC,CO)
Shape(DA,AC,ODC)
Collinear(BDA)
Collinear(BOC)
Cocircular(O,BDC)
Conditions (CDL):
Equal(LengthOfLine(OD),LengthOfLine(OB))
Equal(MeasureOfAngle(BAC),55)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AC,O)
Goal: Value(MeasureOfAngle(COD))
Answer: 70
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AC,O,O)
triangle_property_angle_sum(1,OBD)
triangle_property_angle_sum(1,BAC)
isosceles_triangle_judgment_line_equal(1,OBD)
isosceles_triangle_property_angle_equal(1,OBD)
flat_angle(1,COB)
angle_addition(1,COD,DOB)
Problem: As shown in the diagram, AB=3, AC=2, the center of circle A is A, the tangent to ⊙A is BC. Find the length of line BC.
问题：如图所示，AB=3，AC=2，A是圆A的圆心，BC是圆O的切线。求直线BC的长度。
Construction (CDL):
Shape(CA,AB,BC)
Shape(A)
Cocircular(A,C)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AC),2)
IsCentreOfCircle(A,A)
IsTangentOfCircle(BC,A)
Goal: Value(LengthOfLine(BC))
Answer: sqrt(5)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,BC,A,A)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, ∠BCA=65°, O is the center of ⊙O, the tangent to ⊙O is PA, PB is the tangent to circle O. Find the measure of ∠APB.
问题：如图所示，∠BCA=65°，O是圆O的圆心，PA是⊙O的切线，圆O的切线为PB。求∠APB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(BC,CA,AO,OB)
Shape(OCA,AC)
Shape(BO,OA,OAB)
Shape(OAB,AP,PB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),65)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(APB))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
tangent_of_circle_property_perpendicular(2,PB,O,O)
tangent_of_circle_property_perpendicular(1,PA,O,O)
quadrilateral_property_angle_sum(1,BOAP)
Problem: As shown in the diagram, ∠CDB=30°, the center of circle O is O, EC is the tangent to ⊙O. Find the value of sin(OE).
问题：如图所示，∠CDB=30°，O是⊙O的圆心，圆O的切线为EC。求sin(OE)的值。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OC)
Shape(CO,OM,MC)
Shape(CM,MB,OBC)
Shape(MO,OA,OAD,DM)
Shape(MD,DB,BM)
Shape(BD,ODB)
Shape(OBC,BE,EC)
Collinear(AOMBE)
Collinear(CMD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(EC,O)
Goal: Value(Sin(MeasureOfAngle(OEC)))
Answer: 1/2
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
tangent_of_circle_property_perpendicular(2,EC,O,O)
triangle_property_angle_sum(1,COE)
Problem: As shown in the diagram, PB=2, PT=6, O is the center of circle O, PT is the tangent to circle O. Find the length of line AB.
问题：如图所示，PB=2，PT=6，圆O的圆心为O，PT是圆O的切线。求直线AB的长度。
Construction (CDL):
Shape(TP,PB,OTB)
Shape(OTB,BO,OA,OAT)
Shape(AO,OB,OBA)
Collinear(PBOA)
Cocircular(O,TBA)
Conditions (CDL):
Equal(LengthOfLine(PB),2)
Equal(LengthOfLine(PT),6)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PT,O)
Goal: Value(LengthOfLine(AB))
Answer: 16
Theorem sequence:
line_addition(1,PB,BA)
circle_property_circular_power_tangent_and_segment_line(1,PT,PBA,O)
Problem: As shown in the diagram, CD=4, OD=3, O is the center of ⊙O, the tangent to circle O is CD. Find the value of sin(OC).
问题：如图所示，CD=4，OD=3，O是圆O的圆心，圆O的切线为CD。求sin(OC)的值。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(DO,OB,OBD)
Shape(OBD,BC,CD)
Collinear(AOBC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(LengthOfLine(CD),4)
Equal(LengthOfLine(OD),3)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CD,O)
Goal: Value(Sin(MeasureOfAngle(OCD)))
Answer: 3/5
Theorem sequence:
tangent_of_circle_property_perpendicular(2,CD,O,O)
right_triangle_judgment_angle(1,CDO)
right_triangle_property_pythagorean(1,CDO)
sine_theorem(1,OCD)
Problem: As shown in the diagram, ∠BPA=36°, the center of ⊙O is O, the tangent to circle O is PA. Find the measure of ∠ACB.
问题：如图所示，∠BPA=36°，圆O的圆心为O，PA是圆O的切线。求∠ACB的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CB,BO,OA)
Shape(OCB,BC)
Shape(OB,OBA,AO)
Shape(OBA,BP,PA)
Collinear(OBP)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),36)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 27
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
triangle_property_angle_sum(1,AOP)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠OBA=32°, O is the center of circle O, BA is the tangent to circle O. Find the measure of ∠ADC.
问题：如图所示，∠OBA=32°，⊙O的圆心为O，⊙O的切线为BA。求∠ADC的大小。
Construction (CDL):
Shape(CB,BA,OCA)
Shape(OCA,AM,MC)
Shape(MA,AO,OM)
Shape(ODC,CD)
Shape(DC,CM,MD)
Shape(DM,MO,OA,OAD)
Collinear(BCMO)
Collinear(DMA)
Cocircular(O,DCA)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),32)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 29
Theorem sequence:
tangent_of_circle_property_perpendicular(2,BA,O,O)
triangle_property_angle_sum(1,BAO)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, OC=OB, ∠BCD=50°, O is the center of ⊙O, the tangent to ⊙O is DC. Find the measure of ∠COA.
问题：如图所示，OC=OB，∠BCD=50°，圆O的圆心为O，DC是圆O的切线。求∠COA的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAC,CO,OA)
Shape(OC,CB,BO)
Shape(BC,OCB)
Shape(OCB,CD,DB)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(OC),LengthOfLine(OB))
Equal(MeasureOfAngle(BCD),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 80
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DC,O,O)
angle_addition(1,OCB,BCD)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, OC=OA, ∠APD=∠DPC, ∠DPC=20°, O is the center of ⊙O, PC is the tangent to circle O. Find the measure of ∠CAP.
问题：如图所示，OC=OA，∠APD=∠DPC，∠DPC=20°，⊙O的圆心为O，PC是圆O的切线。求∠CAP的大小。
Construction (CDL):
Shape(OCA,AD,DC)
Shape(DA,AO,OM,MD)
Shape(CD,DM,MC)
Shape(CM,MN,ONC)
Shape(MO,OB,OBN,NM)
Shape(ONC,NP,PC)
Shape(OBN,BP,PN)
Shape(BO,OA,OAB)
Collinear(AOBP)
Collinear(PNMD)
Collinear(ADC)
Collinear(CMO)
Cocircular(O,ABNC)
Conditions (CDL):
Equal(LengthOfLine(OC),LengthOfLine(OA))
Equal(MeasureOfAngle(APD),MeasureOfAngle(DPC))
Equal(MeasureOfAngle(DPC),20)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PC,O)
Goal: Value(MeasureOfAngle(CAP))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PC,O,O)
angle_addition(1,BPN,NPC)
triangle_property_angle_sum(1,COP)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, AB=2, OA=1, OD=3, the center of circle O is O, AB is the diameter of ⊙O, DA is the tangent to ⊙O, CB is parallel to DO. Find the length of line BC.
问题：如图所示，AB=2，OA=1，OD=3，O是圆O的圆心，⊙O的直径为AB，DA是圆O的切线，CB平行于DO。求直线BC的长度。
Construction (CDL):
Shape(ND,DA,ONA)
Shape(ONA,AM,MN)
Shape(OM,MA,AO)
Shape(OCN,NM,MC)
Shape(OBC,CB)
Shape(BC,CM,MO,OB)
Shape(BO,OA,OAB)
Collinear(DNMO)
Collinear(CMA)
Collinear(BOA)
Cocircular(O,BCNA)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(OA),1)
Equal(LengthOfLine(OD),3)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DA,O)
ParallelBetweenLine(CB,DO)
Goal: Value(LengthOfLine(BC))
Answer: 2/3
Theorem sequence:
parallel_property_corresponding_angle(1,OD,BC,A)
tangent_of_circle_property_perpendicular(2,DA,O,O)
diameter_of_circle_property_right_angle(1,BCA,O)
mirror_similar_triangle_judgment_aa(1,ABC,DAO)
mirror_similar_triangle_property_line_ratio(1,ABC,DAO)
mirror_similar_triangle_property_line_ratio(1,CAB,AOD)
Problem: As shown in the diagram, AB=8, AC=BC, OA=OB, the radius of circle O is 3, O is the center of circle O, AC is the tangent to ⊙O. Find the length of line OA.
问题：如图所示，AB=8，AC=BC，OA=OB，⊙O的半径为3，圆O的圆心为O，⊙O的切线为AC。求直线OA的长度。
Construction (CDL):
Shape(CA,AM,OCM)
Shape(OCM,MO,OC)
Shape(BC,ONC,NB)
Shape(ONC,CO,ON)
Shape(OM,OMN,NO)
Collinear(ACB)
Collinear(AMO)
Collinear(BNO)
Cocircular(O,NCM)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AC),LengthOfLine(BC))
Equal(LengthOfLine(OA),LengthOfLine(OB))
Equal(RadiusOfCircle(O),3)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AC,O)
Goal: Value(LengthOfLine(OA))
Answer: 5
Theorem sequence:
tangent_of_circle_property_perpendicular(1,AC,O,O)
radius_of_circle_property_length_equal(1,OC,O)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,OCA)
right_triangle_property_pythagorean(1,OCA)
Problem: As shown in the diagram, ∠CPA=26°, the center of ⊙O is O, PC is the tangent to ⊙O. Find the measure of ∠PAC.
问题：如图所示，∠CPA=26°，⊙O的圆心为O，PC是圆O的切线。求∠PAC的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAC,CA)
Shape(OA,AC,CO)
Shape(OC,OCB,BO)
Shape(OCB,CP,PB)
Collinear(AOBP)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CPA),26)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PC,O)
Goal: Value(MeasureOfAngle(PAC))
Answer: 32
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PC,O,O)
triangle_property_angle_sum(1,OCP)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠CAB=36°, O is the center of circle O, AB is the tangent to circle O. Find the measure of ∠BCA.
问题：如图所示，∠CAB=36°，⊙O的圆心为O，AB是圆O的切线。求∠BCA的大小。
Construction (CDL):
Shape(OCM,MO,OC)
Shape(OMB,BO,OM)
Shape(OB,BC,CO)
Shape(OBC,CB)
Shape(MA,AB,OMB)
Collinear(AMOC)
Cocircular(O,MBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),36)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 27
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,BOA)
arc_property_center_angle(1,OMB,O)
arc_property_circumference_angle_external(1,OMB,C)
Problem: As shown in the diagram, ∠BDO=36°, the center of ⊙O is O, the tangent to circle O is DB. Find the measure of ∠CAB.
问题：如图所示，∠BDO=36°，O是⊙O的圆心，DB是⊙O的切线。求∠CAB的大小。
Construction (CDL):
Shape(BD,DC,OBC)
Shape(MB,OBC,CM)
Shape(OM,MC,CO)
Shape(AB,BM,MA)
Shape(OAB,BA)
Shape(AM,MO,OC,OCA)
Collinear(AMC)
Collinear(OMB)
Collinear(OCD)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BDO),36)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 27
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DB,O,O)
triangle_property_angle_sum(1,BDO)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠APO=28°, the center of circle O is O, the tangent to ⊙O is PA. Find the measure of ∠BCA.
问题：如图所示，∠APO=28°，O是⊙O的圆心，⊙O的切线为PA。求∠BCA的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(MB,BC,CM)
Shape(OCA,AO,OM,MC)
Shape(MO,OA,AM)
Shape(BM,MA,OAB)
Shape(OAB,AP,PB)
Collinear(OMBP)
Collinear(CMA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APO),28)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 31
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
triangle_property_angle_sum(1,OAP)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, AB=8, AH=BH, the radius of ⊙O is 5, O is the center of ⊙O, OH⊥BH. Find the length of line HC.
问题：如图所示，AB=8，AH=BH，⊙O的半径为5，圆O的圆心为O，OH⊥BH。求直线HC的长度。
Construction (CDL):
Shape(OAC,CH,HA)
Shape(OCB,BH,HC)
Shape(OH,HB,BO)
Shape(OB,OBA,AH,HO)
Collinear(AHB)
Collinear(OHC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AH),LengthOfLine(BH))
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OH,BH)
Goal: Value(LengthOfLine(HC))
Answer: 2
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_judgment_angle(1,OHB)
right_triangle_property_pythagorean(1,OHB)
line_addition(1,AH,HB)
line_addition(1,OH,HC)
Problem: As shown in the diagram, AB=8, AO=10, O is the center of ⊙O, AB is the tangent to ⊙O. Find the radius of circle O.
问题：如图所示，AB=8，AO=10，⊙O的圆心为O，⊙O的切线为AB。求圆O的半径。
Construction (CDL):
Shape(AB,OMB,MA)
Shape(OM,OMB,BO)
Shape(OB,OBM,MO)
Collinear(AMO)
Cocircular(O,MB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AO),10)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(RadiusOfCircle(O))
Answer: 6
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
right_triangle_judgment_angle(1,ABO)
right_triangle_property_pythagorean(1,ABO)
radius_of_circle_property_length_equal(1,OB,O)