SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ACO=40°, the center of ⊙O is O, the diameter of circle O is AB, CA is the tangent to circle O. Find the measure of ∠DBA.
问题：如图所示，∠ACO=40°，圆O的圆心为O，圆O的直径为AB，CA是圆O的切线。求∠DBA的大小。
Construction (CDL):
Shape(AC,CD,OAD)
Shape(DO,OA,OAD)
Shape(OD,DB,BO)
Shape(BD,ODB)
Shape(AO,OB,OBA)
Collinear(AOB)
Collinear(CDO)
Cocircular(O,ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ACO),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
triangle_property_angle_sum(1,ODB)
triangle_property_angle_sum(1,ACO)
adjacent_complementary_angle(1,BOD,DOA)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,ODB)
isosceles_triangle_property_angle_equal(1,ODB)
Problem: As shown in the diagram, ∠APO=30°, O is the center of ⊙O, AB is the diameter of circle O, the diameter of ⊙O is CD, the tangent to circle O is PA. Find the measure of ∠OCA.
问题：如图所示，∠APO=30°，O是⊙O的圆心，圆O的直径为AB，⊙O的直径为CD，PA是⊙O的切线。求∠OCA的大小。
Construction (CDL):
Shape(CO,OB,OBC)
Shape(OC,CA,AO)
Shape(AC,OCA)
Shape(BO,OD,ODB)
Shape(DO,OA,OAD)
Shape(AP,PD,OAD)
Collinear(AOB)
Collinear(CODP)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(APO),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CD,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 30
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
triangle_property_angle_sum(1,OCA)
triangle_property_angle_sum(1,OAP)
adjacent_complementary_angle(1,DOA,AOC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, AC=10, ∠BPC=30°, O is the center of ⊙O, the diameter of ⊙O is AC, the tangent to circle O is PB, quadrilateral ABCD is a rectangle. Find the length of line AB.
问题：如图所示，AC=10，∠BPC=30°，O是⊙O的圆心，AC是圆O的直径，圆O的切线为PB，四边形ABCD是矩形。求直线AB的长度。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(DA,AO,OC,CD)
Shape(AB,BO,OA)
Shape(CO,OB,BC)
Shape(BP,PC,OBC)
Collinear(AOCP)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(MeasureOfAngle(BPC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsTangentOfCircle(PB,O)
Rectangle(ABCD)
Goal: Value(LengthOfLine(AB))
Answer: 5*sqrt(3)
Theorem sequence:
line_addition(1,AO,OC)
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)
tangent_of_circle_property_perpendicular(1,PB,O,O)
triangle_property_angle_sum(1,COB)
triangle_property_angle_sum(1,OBP)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
isosceles_triangle_judgment_angle_equal(1,COB)
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠BAC=25°, the center of circle O is O, AB is the diameter of ⊙O, the tangent to circle O is DC. Find the measure of ∠CDB.
问题：如图所示，∠BAC=25°，O是⊙O的圆心，AB是圆O的直径，DC是圆O的切线。求∠CDB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(OA,AC,CO)
Shape(BO,OC,OCB)
Shape(CD,DB,OCB)
Collinear(AOBD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OCB,O)
tangent_of_circle_property_perpendicular(1,DC,O,O)
arc_property_circumference_angle_external(1,OCB,A)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠BCA=40°, PA is the tangent to circle O, PB is the tangent to ⊙O, APBC is a parallelogram. Find the measure of ∠CAB.
问题：如图所示，∠BCA=40°，PA是圆O的切线，PB是⊙O的切线，APBC是▱。求∠CAB的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BA,OAB)
Shape(AD,ODA)
Shape(EB,OBE)
Shape(EC,CD,OED)
Shape(DA,AB,BE,OED)
Collinear(ADC)
Collinear(BEC)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Parallelogram(APBC)
Goal: Value(MeasureOfAngle(CAB))
Answer: 70
Theorem sequence:
tangent_of_circle_property_length_equal(1,PA,PB,O)
parallelogram_property_opposite_line_equal(1,APBC)
parallelogram_property_opposite_line_equal(1,PBCA)
isosceles_triangle_judgment_line_equal(1,CAB)
isosceles_triangle_property_angle_equal(1,CAB)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, OB=3, ∠BPA=30°, O is the center of ⊙O, PA is the tangent to ⊙O. Find the length of line PB.
问题：如图所示，OB=3，∠BPA=30°，O是圆O的圆心，圆O的切线为PA。求直线PB的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Shape(BO,OA,OAB)
Collinear(OBP)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OB),3)
Equal(MeasureOfAngle(BPA),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(PB))
Answer: 3
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
line_addition(1,OB,BP)
sine_theorem(1,OPA)
Problem: As shown in the diagram, ∠BCA=50°, O is the center of ⊙O, the tangent to ⊙O is CB. Find the measure of ∠DOB.
问题：如图所示，∠BCA=50°，O是圆O的圆心，CB是圆O的切线。求∠DOB的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AD,ODA)
Shape(DO,OB,OBD)
Shape(BC,CD,OBD)
Shape(BO,OA,OAB)
Collinear(AOB)
Collinear(ADC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 80
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CB,O,O)
triangle_property_angle_sum(1,ABC)
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠DAB=26°, O is the center of ⊙O, AB is the tangent to circle O. Find the measure of ∠BCA.
问题：如图所示，∠DAB=26°，⊙O的圆心为O，AB是圆O的切线。求∠BCA的大小。
Construction (CDL):
Shape(DA,AB,ODB)
Shape(BC,CO,OB)
Shape(DO,OC,OCD)
Shape(CB,OBC)
Shape(BO,OD,ODB)
Collinear(ADOC)
Cocircular(O,BCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),26)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 32
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,ABO)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, OP=3, ∠ABC=30°, the center of ⊙O is O, the tangent to ⊙O is PA. Find the length of line AP.
问题：如图所示，OP=3，∠ABC=30°，圆O的圆心为O，圆O的切线为PA。求直线AP的长度。
Construction (CDL):
Shape(CP,PA,OCA)
Shape(AO,OC,OCA)
Shape(OA,AB,BC,CO)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(OCP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(OP),3)
Equal(MeasureOfAngle(ABC),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(AP))
Answer: 3*sqrt(3)/2
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
sine_theorem(1,PAO)
Problem: As shown in the diagram, ∠CBA=20°, the center of circle O is O, the diameter of ⊙O is AB, the tangent to ⊙O is PA. Find the measure of ∠APC.
问题：如图所示，∠CBA=20°，O是⊙O的圆心，AB是⊙O的直径，⊙O的切线为PA。求∠APC的大小。
Construction (CDL):
Shape(AP,PC,OAC)
Shape(OC,CB,BO)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AC,CO,OA)
Collinear(PCO)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(APC))
Answer: 50
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
triangle_property_angle_sum(1,APO)
Problem: As shown in the diagram, ∠OBA=32°, O is the center of ⊙O, the tangent to ⊙O is BA. Find the measure of ∠ADC.
问题：如图所示，∠OBA=32°，O是圆O的圆心，BA是圆O的切线。求∠ADC的大小。
Construction (CDL):
Shape(DC,CE,ED)
Shape(OE,EA,AO)
Shape(DE,EO,OA,OAD)
Shape(CD,ODC)
Shape(AE,EC,OCA)
Shape(CB,BA,OCA)
Collinear(BCEO)
Collinear(DEA)
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,AOB)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BAC=25°, O is the center of ⊙O, the diameter of ⊙O is AB, the tangent to ⊙O is DC. Find the measure of ∠CDB.
问题：如图所示，∠BAC=25°，O是圆O的圆心，AB是圆O的直径，圆O的切线为DC。求∠CDB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(CD,DB,OCB)
Collinear(AOBD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DC,O,O)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠ADC=25°, the center of circle O is O, BA is the tangent to circle O. Find the measure of ∠CBA.
问题：如图所示，∠ADC=25°，O是⊙O的圆心，圆O的切线为BA。求∠CBA的大小。
Construction (CDL):
Shape(CB,BA,OCA)
Shape(AO,OC,OCA)
Shape(AD,DC,CO,OA)
Shape(DA,OAD)
Shape(CD,ODC)
Collinear(OCB)
Cocircular(O,ADC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),25)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 40
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,BAO)
Problem: As shown in the diagram, ⌒ODC is congruent to ⌒OCE, ∠BAC=50°, the center of circle O is O, the diameter of ⊙O is BC, AC is the tangent to circle O, the tangent to circle O is FC, the tangent to circle O is FE. Find the measure of ∠CFE.
问题：如图所示，⌒ODC与⌒OCE全等，∠BAC=50°，O是圆O的圆心，BC是⊙O的直径，圆O的切线为AC，FC是⊙O的切线，圆O的切线为FE。求∠CFE的大小。
Construction (CDL):
Shape(DB,OBD)
Shape(CO,OB,BD,ODC)
Shape(DA,AC,ODC)
Shape(BO,OE,OEB)
Shape(EO,OC,OCE)
Shape(CF,FE,OCE)
Collinear(ACF)
Collinear(ADB)
Collinear(BOC)
Cocircular(O,BDCE)
Conditions (CDL):
CongruentBetweenArc(ODC,OCE)
Equal(MeasureOfAngle(BAC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
IsTangentOfCircle(AC,O)
IsTangentOfCircle(FC,O)
IsTangentOfCircle(FE,O)
Goal: Value(MeasureOfAngle(CFE))
Answer: 100
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AC,O,O)
tangent_of_circle_property_perpendicular(1,FC,O,O)
tangent_of_circle_property_perpendicular(2,FE,O,O)
arc_property_center_angle(1,OCE,O)
arc_property_circumference_angle_external(1,ODC,B)
triangle_property_angle_sum(1,CBA)
congruent_arc_property_measure_equal(1,ODC,OCE)
quadrilateral_property_angle_sum(1,EOCF)
Problem: As shown in the diagram, ∠AEB=60°, O is the center of ⊙O, AB is the diameter of circle O, the tangent to ⊙O is PA, the tangent to circle O is PB. Find the measure of ∠BPA.
问题：如图所示，∠AEB=60°，O是⊙O的圆心，AB是圆O的直径，圆O的切线为PA，PB是⊙O的切线。求∠BPA的大小。
Construction (CDL):
Shape(AE,EB,BO,OA)
Shape(AO,OB,OBA)
Shape(EA,OAE)
Shape(BE,OEB)
Shape(BP,PA,OBA)
Cocircular(O,AEB)
Conditions (CDL):
Equal(MeasureOfAngle(AEB),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 60
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,E)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, the measure of arc OBA is 70, O is the center of circle O, AC is the diameter of ⊙O, DA is the tangent to circle O. Find the measure of ∠CDA.
问题：如图所示，⌒OBA的角度为70，O是圆O的圆心，AC是圆O的直径，⊙O的切线为DA。求∠CDA的大小。
Construction (CDL):
Shape(BO,OC,CB)
Shape(AO,OB,OBA)
Shape(BC,OCB)
Shape(BD,DA,OBA)
Shape(CO,OA,OAC)
Collinear(AOC)
Collinear(DBC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfArc(OBA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 55
Theorem sequence:
tangent_of_circle_property_perpendicular(2,DA,O,O)
arc_property_circumference_angle_external(1,OBA,C)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠BPA=40°, the center of circle O is O, the diameter of ⊙O is AC, PA is the tangent to ⊙O, PB is the tangent to ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠BPA=40°，圆O的圆心为O，圆O的直径为AC，PA是圆O的切线，PB是圆O的切线。求∠ACB的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(OC,CB,BO)
Shape(AO,OB,BA)
Shape(BC,OCB)
Shape(AB,OBA)
Shape(BP,PA,OBA)
Collinear(AOC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 70
Theorem sequence:
quadrilateral_property_angle_sum(1,AOBP)
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)
Problem: As shown in the diagram, OA=10, ∠APB=120°, the center of ⊙O is O, the tangent to circle O is PA, PB is the tangent to circle O. Find the length of line AB.
问题：如图所示，OA=10，∠APB=120°，圆O的圆心为O，PA是圆O的切线，圆O的切线为PB。求直线AB的长度。
Construction (CDL):
Shape(AP,PC,OAC)
Shape(CP,PB,OCB)
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(CD,DA,OAC)
Shape(BD,DC,OCB)
Shape(AO,OB,OBA)
Collinear(PCDO)
Collinear(ADB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(OA),10)
Equal(MeasureOfAngle(APB),120)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
tangent_of_circle_property_length_equal(1,PA,PB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
mirror_congruent_triangle_judgment_sss(1,APO,BOP)
mirror_congruent_triangle_property_angle_equal(1,POA,PBO)
mirror_congruent_triangle_property_angle_equal(1,OAP,OPB)
angle_addition(1,APO,OPB)
tangent_of_circle_property_perpendicular(1,PA,O,O)
triangle_property_angle_sum(1,APO)
angle_addition(1,BOP,POA)
cosine_theorem(1,OAB)
Problem: As shown in the diagram, OP=6, ∠POB=90°, C is the center of ⊙C, the tangent to ⊙C is OB, OP is the tangent to circle C. Find the length of line OC.
问题：如图所示，OP=6，∠POB=90°，C是圆C的圆心，OB是圆O的切线，⊙O的切线为OP。求直线OC的长度。
Construction (CDL):
Shape(DC,CP,CPD)
Shape(PO,OD,CPD)
Shape(DO,OB,CDB)
Shape(BC,CD,CDB)
Shape(PC,CB,CBP)
Collinear(CDO)
Cocircular(C,PDB)
Conditions (CDL):
Equal(LengthOfLine(OP),6)
Equal(MeasureOfAngle(POB),90)
IsCentreOfCircle(C,C)
IsTangentOfCircle(OB,C)
IsTangentOfCircle(OP,C)
Goal: Value(LengthOfLine(OC))
Answer: 6*sqrt(2)
Theorem sequence:
radius_of_circle_property_length_equal(1,CP,C)
radius_of_circle_property_length_equal(1,CB,C)
tangent_of_circle_property_perpendicular(1,OP,C,C)
tangent_of_circle_property_length_equal(1,OP,OB,C)
mirror_congruent_triangle_judgment_sss(1,CPO,COB)
mirror_congruent_triangle_property_angle_equal(1,OCP,OBC)
angle_addition(1,DOB,BOP)
angle_addition(1,POB,BOD)
angle_addition(1,BOD,DOP)
triangle_property_angle_sum(1,CPO)
sine_theorem(1,CPO)
sine_theorem(1,POC)
Problem: As shown in the diagram, ∠APB=70°, O is the center of ⊙O, AC is the diameter of circle O, the tangent to circle O is PA, PB is the tangent to ⊙O. Find the measure of ∠PBC.
问题：如图所示，∠APB=70°，⊙O的圆心为O，AC是圆O的直径，圆O的切线为PA，⊙O的切线为PB。求∠PBC的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(AB,BC,CO,OA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AO,OC,OCA)
Collinear(AOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(PBC))
Answer: 145
Theorem sequence:
tangent_of_circle_property_length_equal(1,PA,PB,O)
diameter_of_circle_property_right_angle(1,ABC,O)
isosceles_triangle_judgment_line_equal(1,PBA)
isosceles_triangle_property_angle_equal(1,PBA)
triangle_property_angle_sum(1,APB)
angle_addition(1,PBA,ABC)
Problem: As shown in the diagram, ∠BCD=130°, the center of circle O is O, AB is the diameter of ⊙O, the tangent to circle O is PD. Find the measure of ∠ADP.
问题：如图所示，∠BCD=130°，O是圆O的圆心，圆O的直径为AB，圆O的切线为PD。求∠ADP的大小。
Construction (CDL):
Shape(DP,PA,ODA)
Shape(AD,ODA)
Shape(DA,AO,OB,BD)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Shape(BC,CD,DB)
Collinear(PAOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),130)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PD,O)
Goal: Value(MeasureOfAngle(ADP))
Answer: 40
Theorem sequence:
angle_addition(1,BDA,ADP)
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,ODB,C)
circle_property_angle_of_osculation(1,ODB,P)
Problem: As shown in the diagram, ∠BPA=40°, the center of ⊙O is O, AB is the diameter of ⊙O, the tangent to ⊙O is PB. Find the measure of ∠ADC.
问题：如图所示，∠BPA=40°，圆O的圆心为O，圆O的直径为AB，PB是圆O的切线。求∠ADC的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AE,EC,CA)
Shape(CE,ED,DC)
Shape(EA,AO,OB,BE)
Shape(AC,OCA)
Shape(CD,ODC)
Shape(BP,PC,ODC,OBD)
Shape(DE,EB,OBD)
Collinear(BEC)
Collinear(AED)
Collinear(ACP)
Collinear(AOB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
triangle_property_angle_sum(1,ABP)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BAO=40°, O is the center of circle O, the tangent to circle O is AB. Find the measure of ∠BCO.
问题：如图所示，∠BAO=40°，O是⊙O的圆心，AB是⊙O的切线。求∠BCO的大小。
Construction (CDL):
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BA,AC,OBC)
Shape(BO,OC,OCB)
Collinear(OCA)
Cocircular(O,BC)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 65
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
tangent_of_circle_property_perpendicular(1,AB,O,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,COB)
triangle_property_angle_sum(1,OBA)
Problem: As shown in the diagram, OA=3, OP=5, the center of circle O is O, the diameter of circle O is AB, the tangent to ⊙O is PA. Find the length of line PA.
问题：如图所示，OA=3，OP=5，O是⊙O的圆心，圆O的直径为AB，圆O的切线为PA。求直线PA的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Shape(BO,OA,OAB)
Collinear(OBP)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OA),3)
Equal(LengthOfLine(OP),5)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(PA))
Answer: 4
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
right_triangle_judgment_angle(1,PAO)
right_triangle_property_pythagorean(1,PAO)
Problem: As shown in the diagram, ∠APB=50°, the center of circle O is O, PA is the tangent to circle O, PB is the tangent to ⊙O. Find the measure of ∠BAP.
问题：如图所示，∠APB=50°，⊙O的圆心为O，⊙O的切线为PA，PB是⊙O的切线。求∠BAP的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BA,OAB)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(APB),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BAP))
Answer: 65
Theorem sequence:
tangent_of_circle_property_length_equal(1,PA,PB,O)
isosceles_triangle_judgment_line_equal(1,PBA)
isosceles_triangle_property_angle_equal(1,PBA)
triangle_property_angle_sum(1,APB)
Problem: As shown in the diagram, ∠AEB=65°, O is the center of circle O, the tangent to ⊙O is PA, PB is the tangent to circle O. Find the measure of ∠BPA.
问题：如图所示，∠AEB=65°，⊙O的圆心为O，圆O的切线为PA，⊙O的切线为PB。求∠BPA的大小。
Construction (CDL):
Shape(EA,OAE)
Shape(BE,OEB)
Shape(AE,EB,BO,OA)
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Cocircular(O,AEB)
Conditions (CDL):
Equal(MeasureOfAngle(AEB),65)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 50
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,E)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, ∠CAB=25°, O is the center of ⊙O, AB is the diameter of ⊙O, DC is the tangent to ⊙O. Find the measure of ∠BDC.
问题：如图所示，∠CAB=25°，O是⊙O的圆心，AB是圆O的直径，⊙O的切线为DC。求∠BDC的大小。
Construction (CDL):
Shape(BD,DC,OBC)
Shape(CB,OBC)
Shape(AO,OB,BC,CA)
Shape(AC,OCA)
Shape(BO,OA,OAB)
Collinear(AOBD)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,ABC)
circle_property_circular_power_tangent_and_segment_angle(2,DC,DBA,O)
Problem: As shown in the diagram, ∠BPA=40°, O is the center of circle O, PA is the tangent to circle O, PB is the tangent to circle O. Find the measure of ∠BDA.
问题：如图所示，∠BPA=40°，⊙O的圆心为O，⊙O的切线为PA，PB是圆O的切线。求∠BDA的大小。
Construction (CDL):
Shape(AO,OB,BD,DA)
Shape(BO,OA,OAB)
Shape(AD,ODA)
Shape(DB,OBD)
Shape(BP,PA,ODA,OBD)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 110
Theorem sequence:
round_angle(1,AOB,BOA)
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,D)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, ∠OPA=40°, the center of ⊙O is O, the diameter of ⊙O is AB, the tangent to circle O is PA. Find the measure of ∠ABC.
问题：如图所示，∠OPA=40°，O是圆O的圆心，AB是圆O的直径，PA是圆O的切线。求∠ABC的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CP,PA,OCA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Collinear(OCP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OPA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,AOP)
Problem: As shown in the diagram, ∠BAC=30°, the radius of ⊙O is 2, O is the center of ⊙O, the diameter of circle O is AB, the tangent to ⊙O is DC. Find the length of line BD.
问题：如图所示，∠BAC=30°，圆O的半径为2，O是⊙O的圆心，AB是⊙O的直径，DC是圆O的切线。求直线BD的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,AC,CO)
Shape(BO,OC,OCB)
Shape(CA,OAC)
Shape(CD,DB,OCB)
Collinear(AOBD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(LengthOfLine(BD))
Answer: 2
Theorem sequence:
line_addition(1,OB,BD)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
tangent_of_circle_property_perpendicular(1,DC,O,O)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
triangle_property_angle_sum(1,OCD)
sine_theorem(1,OCD)
Problem: As shown in the diagram, ∠CPA=20°, the center of ⊙O is O, AB is the diameter of circle O, the tangent to circle O is PA. Find the measure of ∠ABC.
问题：如图所示，∠CPA=20°，⊙O的圆心为O，圆O的直径为AB，PA是圆O的切线。求∠ABC的大小。
Construction (CDL):
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CP,PA,OCA)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(OCP)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CPA),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 35
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,AOP)