SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=10, AC=6, ∠BCA=90°. Find the length of line BC.
问题：如图所示，AB=10，AC=6，∠BCA=90°。求直线BC的长度。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),6)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
sine_theorem(1,ABC)
cosine_theorem(1,BCA)
Problem: As shown in the diagram, PC=3, ∠CAB=30°, the center of circle O is O, the diameter of ⊙O is AB, PC is the tangent to circle O. Find the radius of ⊙O.
问题：如图所示，PC=3，∠CAB=30°，O是圆O的圆心，AB是圆O的直径，圆O的切线为PC。求⊙O的半径。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(CO,OB,OBC)
Shape(BP,PC,OBC)
Shape(BO,OA,OAB)
Collinear(AOBP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(PC),3)
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PC,O)
Goal: Value(RadiusOfCircle(O))
Answer: sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
tangent_of_circle_property_perpendicular(2,PC,O,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,COP)
sine_theorem(1,COP)
Problem: As shown in the diagram, ∠ADC=90°, ∠BAD=60°, the center of circle O is O, the diameter of ⊙O is AB, the tangent to circle O is DC. Find the measure of ∠DCB.
问题：如图所示，∠ADC=90°，∠BAD=60°，O是圆O的圆心，圆O的直径为AB，DC是⊙O的切线。求∠DCB的大小。
Construction (CDL):
Shape(BO,OC,CB)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Shape(CO,OA,AE,OEC)
Shape(ED,DC,OEC)
Shape(EA,OAE)
Collinear(AOB)
Collinear(AED)
Cocircular(O,AECB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),90)
Equal(MeasureOfAngle(BAD),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 150
Theorem sequence:
tangent_of_circle_property_perpendicular(2,DC,O,O)
quadrilateral_property_angle_sum(1,OADC)
adjacent_complementary_angle(1,BOC,COA)
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)
angle_addition(1,DCO,OCB)
Problem: As shown in the diagram, ∠CDA=90°, ∠DAB=70°, O is the center of ⊙O, AB is the diameter of ⊙O, the tangent to circle O is DC. Find the measure of ∠ACD.
问题：如图所示，∠CDA=90°，∠DAB=70°，圆O的圆心为O，AB是圆O的直径，圆O的切线为DC。求∠ACD的大小。
Construction (CDL):
Shape(AO,OC,CA)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Shape(CD,DE,OCE)
Shape(AE,OEA)
Shape(EA,AC,OCE)
Collinear(AED)
Collinear(AOB)
Cocircular(O,ABCE)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),90)
Equal(MeasureOfAngle(DAB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 55
Theorem sequence:
angle_addition(1,OCA,ACD)
angle_addition(1,CAO,OAE)
angle_addition(1,EAO,OAC)
angle_addition(1,OAC,CAE)
tangent_of_circle_property_perpendicular(1,DC,O,O)
arc_property_center_angle(1,OCA,O)
circle_property_angle_of_osculation(1,OCA,D)
triangle_property_angle_sum(1,AOC)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠BAN=50°, O is the center of circle O, the diameter of circle O is AD, the tangent to circle O is NA. Find the measure of ∠BCA.
问题：如图所示，∠BAN=50°，O是⊙O的圆心，圆O的直径为AD，圆O的切线为NA。求∠BCA的大小。
Construction (CDL):
Shape(CA,AO,OE,EC)
Shape(AC,OCA)
Shape(EO,OA,AB,BE)
Shape(BA,OAB)
Shape(CE,ED,ODC)
Shape(DE,EB,OBD)
Shape(BA,AN)
Collinear(AOED)
Collinear(CEB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(BAN),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
IsTangentOfCircle(NA,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
circle_property_angle_of_osculation(1,OAB,N)
Problem: As shown in the diagram, ∠AED=40°, O is the center of ⊙O, the diameter of circle O is AB, the tangent to circle O is CB. Find the measure of ∠BCA.
问题：如图所示，∠AED=40°，圆O的圆心为O，AB是圆O的直径，圆O的切线为CB。求∠BCA的大小。
Construction (CDL):
Shape(AE,EF,FO,OA)
Shape(AO,OF,FD,DA)
Shape(EA,OAE)
Shape(AD,ODA)
Shape(BF,FE,OEB)
Shape(DF,FB,BD)
Shape(DB,OBD)
Shape(BC,CD,OBD)
Collinear(AOFB)
Collinear(EFD)
Collinear(ADC)
Cocircular(O,AEBD)
Conditions (CDL):
Equal(MeasureOfAngle(AED),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CB,O,O)
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,ODA,E)
arc_property_circumference_angle_external(1,ODA,B)
triangle_property_angle_sum(1,DAB)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠ABC=60°, the radius of circle O is 1, the center of circle O is O, DC is the diameter of circle O, PA is the tangent to circle O. Find the length of line PA.
问题：如图所示，∠ABC=60°，⊙O的半径为1，⊙O的圆心为O，⊙O的直径为DC，PA是圆O的切线。求直线PA的长度。
Construction (CDL):
Shape(AP,PD,OAD)
Shape(DA,OAD)
Shape(AD,DE,EA)
Shape(AE,EO,OA)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(CO,OE,EB,BC)
Shape(BE,ED,ODB)
Collinear(PDEOC)
Collinear(AEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DC,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(PA))
Answer: sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
diameter_of_circle_property_right_angle(1,CAD,O)
tangent_of_circle_property_perpendicular(1,PA,O,O)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,APO)
sine_theorem(1,APO)
Problem: As shown in the diagram, ∠CAB=31°, O is the center of circle O, DC is the tangent to circle O. Find the measure of ∠BDC.
问题：如图所示，∠CAB=31°，圆O的圆心为O，⊙O的切线为DC。求∠BDC的大小。
Construction (CDL):
Shape(CO,OE,EC)
Shape(CE,EB,OBC)
Shape(BD,DC,OBC)
Shape(BE,EA,AB)
Shape(BA,OAB)
Shape(AE,EO,OC,OCA)
Collinear(OEBD)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),31)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 28
Theorem sequence:
tangent_of_circle_property_perpendicular(2,DC,O,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,COD)
Problem: As shown in the diagram, OC=4, ∠CDO=90°, ∠DOC=45°. Find the length of line CD.
问题：如图所示，OC=4，∠CDO=90°，∠DOC=45°。求直线CD的长度。
Construction (CDL):
Shape(DO,OC,CD)
Conditions (CDL):
Equal(LengthOfLine(OC),4)
Equal(MeasureOfAngle(CDO),90)
Equal(MeasureOfAngle(DOC),45)
Goal: Value(LengthOfLine(CD))
Answer: 2*sqrt(2)
Theorem sequence:
sine_theorem(1,CDO)
Problem: As shown in the diagram, ∠BPA=50°, the center of ⊙O is O, the tangent to circle O is PA, the tangent to ⊙O is PB. Find the measure of ∠ACB.
问题：如图所示，∠BPA=50°，O是⊙O的圆心，⊙O的切线为PA，PB是圆O的切线。求∠ACB的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Shape(AC,CB,BO,OA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 65
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, ∠OBA=28°, O is the center of circle O, BD is the diameter of ⊙O, the tangent to ⊙O is CA. Find the measure of ∠ACO.
问题：如图所示，∠OBA=28°，O是圆O的圆心，圆O的直径为BD，CA是⊙O的切线。求∠ACO的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(AB,OBA)
Shape(DO,OA,OAD)
Shape(AC,CD,OAD)
Shape(BO,OD,ODB)
Collinear(BODC)
Cocircular(O,ADB)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),28)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(ACO))
Answer: 34
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,B)
triangle_property_angle_sum(1,OAC)
Problem: As shown in the diagram, PA=2, O is the center of ⊙O, the tangent to ⊙O is EA, EC is the tangent to circle O, the tangent to circle O is FB, FC is the tangent to circle O, PA is the tangent to ⊙O, PB is the tangent to circle O. Find the perimeter of △EPF.
问题：如图所示，PA=2，⊙O的圆心为O，EA是⊙O的切线，EC是⊙O的切线，⊙O的切线为FB，⊙O的切线为FC，PA是⊙O的切线，PB是⊙O的切线。求△EPF的周长。
Construction (CDL):
Shape(EP,PF,FC,CE)
Shape(AE,EC,OAC)
Shape(CF,FB,OCB)
Collinear(PEA)
Collinear(PFB)
Collinear(ECF)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(PA),2)
IsCentreOfCircle(O,O)
IsTangentOfCircle(EA,O)
IsTangentOfCircle(EC,O)
IsTangentOfCircle(FB,O)
IsTangentOfCircle(FC,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(PerimeterOfTriangle(EPF))
Answer: 4
Theorem sequence:
tangent_of_circle_property_length_equal(1,PA,PB,O)
tangent_of_circle_property_length_equal(1,EC,EA,O)
tangent_of_circle_property_length_equal(1,FC,FB,O)
line_addition(1,PE,EA)
line_addition(1,PF,FB)
line_addition(1,EC,CF)
triangle_perimeter_formula(1,EPF)
Problem: As shown in the diagram, AB=10, BC=7, CD=8, the center of ⊙O is O, AE is the tangent to circle O, the tangent to circle O is AH, the tangent to ⊙O is BE, BF is the tangent to ⊙O, CF is the tangent to ⊙O, the tangent to ⊙O is CG, DG is the tangent to ⊙O, the tangent to circle O is DH. Find the length of line AD.
问题：如图所示，AB=10，BC=7，CD=8，⊙O的圆心为O，AE是圆O的切线，圆O的切线为AH，⊙O的切线为BE，圆O的切线为BF，CF是圆O的切线，圆O的切线为CG，DG是圆O的切线，DH是⊙O的切线。求直线AD的长度。
Construction (CDL):
Shape(EB,BF,OEF)
Shape(FC,CG,OFG)
Shape(GD,DH,OGH)
Shape(HA,AE,OHE)
Collinear(AEB)
Collinear(BFC)
Collinear(CGD)
Collinear(DHA)
Cocircular(O,EFGH)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),7)
Equal(LengthOfLine(CD),8)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AE,O)
IsTangentOfCircle(AH,O)
IsTangentOfCircle(BE,O)
IsTangentOfCircle(BF,O)
IsTangentOfCircle(CF,O)
IsTangentOfCircle(CG,O)
IsTangentOfCircle(DG,O)
IsTangentOfCircle(DH,O)
Goal: Value(LengthOfLine(AD))
Answer: 11
Theorem sequence:
tangent_of_circle_property_length_equal(1,DG,DH,O)
tangent_of_circle_property_length_equal(1,AH,AE,O)
tangent_of_circle_property_length_equal(1,BE,BF,O)
tangent_of_circle_property_length_equal(1,CF,CG,O)
line_addition(1,AE,EB)
line_addition(1,BF,FC)
line_addition(1,CG,GD)
line_addition(1,DH,HA)
Problem: As shown in the diagram, ∠BCA=26°, O is the center of ⊙O, CD is the diameter of circle O, the tangent to circle O is AB. Find the measure of ∠DAB.
问题：如图所示，∠BCA=26°，⊙O的圆心为O，⊙O的直径为CD，圆O的切线为AB。求∠DAB的大小。
Construction (CDL):
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BO,OD,ODB)
Shape(DA,AB,ODB)
Shape(DO,OC,OCD)
Collinear(ADOC)
Cocircular(O,DBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),26)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 38
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
triangle_property_angle_sum(1,BOA)
Problem: As shown in the diagram, ∠DAB=40°, O is the center of circle O, CD is the diameter of circle O, the tangent to circle O is AB. Find the measure of ∠BCA.
问题：如图所示，∠DAB=40°，O是⊙O的圆心，⊙O的直径为CD，圆O的切线为AB。求∠BCA的大小。
Construction (CDL):
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BO,OD,ODB)
Shape(DA,AB,ODB)
Shape(DO,OC,OCD)
Collinear(ADOC)
Cocircular(O,DBC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
triangle_property_angle_sum(1,BOA)
Problem: As shown in the diagram, AD=6, ∠DAF=∠FAE, ∠EAC=60°, O is the center of ⊙O, AD is the tangent to circle O, AE is the tangent to ⊙O, triangle BAC is a right triangle. Find the length of line OD.
问题：如图所示，AD=6，∠DAF=∠FAE，∠EAC=60°，⊙O的圆心为O，⊙O的切线为AD，AE是⊙O的切线，∠BAC是直角△BAC的直角。求直线OD的长度。
Construction (CDL):
Shape(BE,EA,AC,CB)
Shape(DO,OE,OED)
Shape(FA,AE,OFE)
Shape(DA,AF,ODF)
Shape(EO,OF,OFE)
Shape(FO,OD,ODF)
Collinear(OFA)
Collinear(DAC)
Collinear(BEA)
Cocircular(O,DFE)
Conditions (CDL):
Equal(LengthOfLine(AD),6)
Equal(MeasureOfAngle(DAF),MeasureOfAngle(FAE))
Equal(MeasureOfAngle(EAC),60)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AD,O)
IsTangentOfCircle(AE,O)
RightTriangle(BAC)
Goal: Value(LengthOfLine(OD))
Answer: 6*sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(1,AD,O,O)
angle_addition(1,DAE,EAC)
flat_angle(1,DAC)
angle_addition(1,DAO,OAE)
triangle_property_angle_sum(1,ODA)
sine_theorem(1,DAO)
Problem: As shown in the diagram, AB=12, OA=5, the center of ⊙O is O, BA is the tangent to circle O. Find the length of line BC.
问题：如图所示，AB=12，OA=5，O是圆O的圆心，圆O的切线为BA。求直线BC的长度。
Construction (CDL):
Shape(AB,BC,OAC)
Shape(AO,CO,OCA)
Shape(CO,OA,OAC)
Collinear(OCB)
Cocircular(O,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(OA),5)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,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)
tangent_of_circle_property_perpendicular(1,BA,O,O)
right_triangle_judgment_angle(1,OAB)
right_triangle_property_pythagorean(1,OAB)
line_addition(1,OC,CB)
Problem: As shown in the diagram, ∠ABE=29°, the center of ⊙O is O, DC is the tangent to ⊙O. Find the measure of ∠CDA.
问题：如图所示，∠ABE=29°，O是圆O的圆心，圆O的切线为DC。求∠CDA的大小。
Construction (CDL):
Shape(CD,DA,OCA)
Shape(BA,OAB)
Shape(AO,OC,OCA)
Shape(CE,EB,OBC)
Shape(OE,EC,CO)
Shape(AB,BE,EO,OA)
Collinear(DAOE)
Collinear(BEC)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),29)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 32
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DC,O,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠EAB=65°, the center of circle O is O, G bisects segment AC, DB is the tangent to circle O. Find the measure of ∠BDC.
问题：如图所示，∠EAB=65°，⊙O的圆心为O，G平分线段AC，DB是⊙O的切线。求∠BDC的大小。
Construction (CDL):
Shape(AE,OEA)
Shape(AG,GE,EA)
Shape(EG,GF,FE)
Shape(EF,FC,OCE)
Shape(BA,OAB)
Shape(AB,BO,OG,GA)
Shape(FG,GO,OB,BF)
Shape(CF,FB,OBC)
Shape(BD,DC,OBC)
Collinear(AGFCD)
Collinear(BFE)
Collinear(OGE)
Cocircular(O,BCEA)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),65)
IsCentreOfCircle(O,O)
IsMidpointOfLine(G,AC)
IsTangentOfCircle(DB,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBE,O)
arc_property_circumference_angle_external(1,OBE,A)
tangent_of_circle_property_perpendicular(1,DB,O,O)
triangle_property_angle_sum(1,EOB)
circle_property_chord_perpendicular_bisect_chord(2,O,OG,CA)
vertical_angle(1,AFE,CFB)
vertical_angle(1,OGA,EGF)
triangle_property_angle_sum(1,EGF)
angle_addition(1,OBF,FBD)
triangle_property_angle_sum(1,FBD)
Problem: As shown in the diagram, OA=5, OB=5, OC=5, ∠CAO=30°, O is the center of circle O, the tangent to circle O is PC. Find the length of line CP.
问题：如图所示，OA=5，OB=5，OC=5，∠CAO=30°，圆O的圆心为O，圆O的切线为PC。求直线CP的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,OBC)
Shape(BP,PC,OBC)
Shape(BO,OA,OAB)
Collinear(AOBP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(OA),5)
Equal(LengthOfLine(OB),5)
Equal(LengthOfLine(OC),5)
Equal(MeasureOfAngle(CAO),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PC,O)
Goal: Value(LengthOfLine(CP))
Answer: 5*sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PC,O,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
angle_addition(1,PCO,OCA)
cosine_theorem(1,AOC)
triangle_property_angle_sum(1,CAP)
isosceles_triangle_judgment_angle_equal(1,CAP)
Problem: As shown in the diagram, ∠APC=40°, the center of circle O is O, the tangent to circle O is PA. Find the measure of ∠CBO.
问题：如图所示，∠APC=40°，⊙O的圆心为O，PA是⊙O的切线。求∠CBO的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(OC,CB,BO)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Shape(AP,PC,OAC)
Collinear(AOB)
Collinear(PCO)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(APC),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(CBO))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
triangle_property_angle_sum(1,APO)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠CDP=70°, ∠PAB=30°, the center of circle O is O, AP is the tangent to ⊙O. Find the measure of ∠OBC.
问题：如图所示，∠CDP=70°，∠PAB=30°，⊙O的圆心为O，圆O的切线为AP。求∠OBC的大小。
Construction (CDL):
Shape(PD,ODP)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(PO,OC,CD,DP)
Shape(OB,BC,CO)
Shape(BO,OP,OPB)
Shape(PA,AB,OPB)
Collinear(ABO)
Cocircular(O,CDPB)
Conditions (CDL):
Equal(MeasureOfAngle(CDP),70)
Equal(MeasureOfAngle(PAB),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AP,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 50
Theorem sequence:
tangent_of_circle_property_perpendicular(1,AP,O,O)
triangle_property_angle_sum(1,PAO)
arc_property_circumference_angle_external(1,OPC,D)
arc_property_center_angle(1,OPC,O)
angle_addition(1,COB,BOP)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠DCA=50°, O is the center of ⊙O, the tangent to ⊙O is CA. Find the measure of ∠ABO.
问题：如图所示，∠DCA=50°，圆O的圆心为O，CA是⊙O的切线。求∠ABO的大小。
Construction (CDL):
Shape(DO,OB,OBD)
Shape(BO,OA,AB)
Shape(AO,OD,ODA)
Shape(DC,CA,ODA)
Shape(BA,OAB)
Collinear(BODC)
Cocircular(O,DAB)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 20
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
tangent_of_circle_property_perpendicular(2,CA,O,O)
triangle_property_angle_sum(1,AOC)
flat_angle(1,BOC)
angle_addition(1,BOA,AOD)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,BOA)
Problem: As shown in the diagram, ∠EBC=32°, O is the center of ⊙O, the tangent to ⊙O is PA. Find the measure of ∠CPA.
问题：如图所示，∠EBC=32°，O是圆O的圆心，PA是⊙O的切线。求∠CPA的大小。
Construction (CDL):
Shape(EA,AO,OE)
Shape(AE,EC,OCA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BE,EO,OA,OAB)
Shape(CP,PA,OCA)
Collinear(PCEO)
Collinear(AEB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(EBC),32)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(CPA))
Answer: 26
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
triangle_property_angle_sum(1,PAO)
Problem: As shown in the diagram, ∠BCD=40°, O is the center of ⊙O, CB is the tangent to ⊙O. Find the measure of ∠ABO.
问题：如图所示，∠BCD=40°，O是圆O的圆心，圆O的切线为CB。求∠ABO的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,ODA)
Shape(BA,OAB)
Shape(DO,OB,OBD)
Shape(BC,CD,OBD)
Collinear(AODC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 25
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
tangent_of_circle_property_perpendicular(1,CB,O,O)
triangle_property_angle_sum(1,COB)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,A)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
Problem: As shown in the diagram, ∠CBO=21°, the center of circle O is O, DA is the tangent to ⊙O. Find the measure of ∠ADC.
问题：如图所示，∠CBO=21°，圆O的圆心为O，圆O的切线为DA。求∠ADC的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(BO,OC,CB)
Shape(CO,OA,OAC)
Shape(AD,DC,OAC)
Shape(AO,OB,OBA)
Collinear(BOA)
Collinear(OCD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),21)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 48
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
tangent_of_circle_property_perpendicular(1,DA,O,O)
arc_property_center_angle(1,OAC,O)
triangle_property_angle_sum(1,DOA)
Problem: As shown in the diagram, ∠ECA=25°, O is the center of circle O, the tangent to circle O is DB, the tangent to ⊙O is DC. Find the measure of ∠CDB.
问题：如图所示，∠ECA=25°，圆O的圆心为O，DB是圆O的切线，⊙O的切线为DC。求∠CDB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(AC,CO,OA)
Shape(BO,OC,OCB)
Shape(CA,OAC)
Shape(EC,OAC)
Shape(CD,DB,OCB)
Collinear(AOB)
Collinear(ECD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ECA),25)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 50
Theorem sequence:
flat_angle(1,ECD)
angle_addition(1,ECA,ACD)
tangent_of_circle_property_perpendicular(1,DC,O,O)
tangent_of_circle_property_perpendicular(2,DB,O,O)
angle_addition(1,ACO,OCD)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
quadrilateral_property_angle_sum(1,BOCD)
Problem: As shown in the diagram, ∠BAC=40°, O is the center of circle O, AB is the tangent to ⊙O. Find the measure of ∠CBA.
问题：如图所示，∠BAC=40°，⊙O的圆心为O，AB是⊙O的切线。求∠CBA的大小。
Construction (CDL):
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BA,AC,OBC)
Shape(CO,OB,OCB)
Collinear(OCA)
Cocircular(O,BC)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(1,AB,O,O)
triangle_property_angle_sum(1,OBA)
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)
angle_addition(1,OBC,CBA)
Problem: As shown in the diagram, ∠CAO=35°, the center of circle O is O, BA is the tangent to ⊙O. Find the measure of ∠ABD.
问题：如图所示，∠CAO=35°，O是⊙O的圆心，⊙O的切线为BA。求∠ABD的大小。
Construction (CDL):
Shape(CO,OD,ODC)
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(DO,OA,OAD)
Shape(AB,BD,OAD)
Collinear(CODB)
Cocircular(O,CA)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),35)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 20
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
tangent_of_circle_property_perpendicular(1,BA,O,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
angle_addition(1,CAO,OAB)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠BDA=36°, O is the center of circle O, the tangent to ⊙O is DA. Find the measure of ∠ACB.
问题：如图所示，∠BDA=36°，O是⊙O的圆心，DA是⊙O的切线。求∠ACB的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(BO,OA,AC,CB)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BD,DA,OBA)
Collinear(DBO)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(BDA),36)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 27
Theorem sequence:
tangent_of_circle_property_perpendicular(2,DA,O,O)
triangle_property_angle_sum(1,ODA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠EAB=28°, the center of circle O is O, the tangent to ⊙O is DC. Find the measure of ∠BDC.
问题：如图所示，∠EAB=28°，⊙O的圆心为O，圆O的切线为DC。求∠BDC的大小。
Construction (CDL):
Shape(CO,OE,EC)
Shape(BE,EA,AB)
Shape(BA,OAB)
Shape(CE,EB,OBC)
Shape(AE,EO,OC,OCA)
Shape(BD,DC,OBC)
Collinear(OEBD)
Collinear(CEA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),28)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 34
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
tangent_of_circle_property_perpendicular(2,DC,O,O)
triangle_property_angle_sum(1,COD)