SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BOA=60°, O is the center of circle O, AB is the tangent to ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠BOA=60°，圆O的圆心为O，AB是⊙O的切线。求∠CAB的大小。
Construction (CDL):
Shape(OCB,BO,OC)
Shape(CA,AB,OCB)
Shape(OBC,CO,OB)
Collinear(ACO)
Cocircular(O,CB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,BOA)
Problem: As shown in the diagram, the diameter of circle O is 2, ∠OCD=30°, O is the center of ⊙O, AB is the diameter of ⊙O, the tangent to circle O is CD. Find the length of line AD.
问题：如图所示，圆O的直径为2，∠OCD=30°，⊙O的圆心为O，⊙O的直径为AB，圆O的切线为CD。求直线AD的长度。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OBD,DO,OB)
Shape(ODA,AD)
Shape(OAB,BO,OA)
Shape(CD,OBD,BC)
Collinear(AOBC)
Cocircular(O,DAB)
Conditions (CDL):
Equal(DiameterOfCircle(O),2)
Equal(MeasureOfAngle(OCD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Goal: Value(LengthOfLine(AD))
Answer: sqrt(3)
Theorem sequence:
adjacent_complementary_angle(1,AOD,DOB)
triangle_property_angle_sum(1,DAO)
triangle_property_angle_sum(1,DOC)
sine_theorem(1,AOD)
sine_theorem(1,ODA)
tangent_of_circle_property_perpendicular(2,CD,O,O)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, ∠CPA=40°, O is the center of ⊙O, the diameter of ⊙O is AB, the tangent to circle O is PA. Find the measure of ∠OBC.
问题：如图所示，∠CPA=40°，O是圆O的圆心，⊙O的直径为AB，PA是圆O的切线。求∠OBC的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(BC,CO,OB)
Shape(OBC,CB)
Shape(CP,PA,OCA)
Shape(OAB,BO,OA)
Collinear(AOB)
Collinear(OCP)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(CPA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 25
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
triangle_property_angle_sum(1,AOP)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠ABC=30°, the radius of circle O is 2, the center of ⊙O is O, the tangent to ⊙O is BA. Find the length of line AB.
问题：如图所示，∠ABC=30°，圆O的半径为2，O是圆O的圆心，BA是圆O的切线。求直线AB的长度。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCA,AO,OC)
Shape(OAC,AB,BC)
Collinear(OCB)
Cocircular(O,AC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(1,BA,O,O)
triangle_property_angle_sum(1,OAB)
radius_of_circle_property_length_equal(1,OA,O)
sine_theorem(1,ABO)
Problem: As shown in the diagram, ∠ACD=70°, the center of circle O is O, the diameter of ⊙O is BA, CA is the tangent to circle O. Find the measure of ∠DOA.
问题：如图所示，∠ACD=70°，O是圆O的圆心，BA是⊙O的直径，⊙O的切线为CA。求∠DOA的大小。
Construction (CDL):
Shape(OAD,DO,OA)
Shape(ODB,BD)
Shape(DB,BO,OD)
Shape(OBA,AO,OB)
Shape(AC,CD,OAD)
Collinear(BOA)
Collinear(BDC)
Cocircular(O,BAD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(DOA))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
triangle_property_angle_sum(1,BAC)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_center_angle(1,OAD,O)
Problem: As shown in the diagram, ∠AEB=50°, O is the center of ⊙O, PA is the tangent to circle O, PB is the tangent to circle O. Find the measure of ∠BPA.
问题：如图所示，∠AEB=50°，O是圆O的圆心，PA是圆O的切线，PB是⊙O的切线。求∠BPA的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAE,EA)
Shape(OEB,BE)
Shape(AE,EB,BO,OA)
Shape(PA,OBA,BP)
Cocircular(O,AEB)
Conditions (CDL):
Equal(MeasureOfAngle(AEB),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 80
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_circumference_angle_external(1,OBA,E)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠ABO=20°, the center of circle O is O, the diameter of ⊙O is BD, CA is the tangent to ⊙O. Find the measure of ∠DCA.
问题：如图所示，∠ABO=20°，⊙O的圆心为O，⊙O的直径为BD，CA是圆O的切线。求∠DCA的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(CA,ODA,DC)
Shape(OAB,BA)
Shape(OBD,DO,OB)
Shape(AO,OD,ODA)
Collinear(BODC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(DCA))
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,ABO)
triangle_property_angle_sum(1,CAO)
tangent_of_circle_property_perpendicular(2,CA,O,O)
adjacent_complementary_angle(1,BOA,AOD)
Problem: As shown in the diagram, ∠AEB=60°, the center of ⊙O is O, PA is the tangent to circle O, PB is the tangent to ⊙O. Find the measure of ∠BPA.
问题：如图所示，∠AEB=60°，O是⊙O的圆心，PA是⊙O的切线，⊙O的切线为PB。求∠BPA的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAE,EA)
Shape(OEB,BE)
Shape(AE,EB,BO,OA)
Shape(PA,OBA,BP)
Cocircular(O,AEB)
Conditions (CDL):
Equal(MeasureOfAngle(AEB),60)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,E)
arc_property_center_angle(1,OBA,O)
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, ∠CAO=30°, the radius of circle O is 5, the center of ⊙O is O, PC is the tangent to circle O. Find the length of line BP.
问题：如图所示，∠CAO=30°，圆O的半径为5，O是圆O的圆心，⊙O的切线为PC。求直线BP的长度。
Construction (CDL):
Shape(CA,AO,OC)
Shape(OBC,CO,OB)
Shape(OCA,AC)
Shape(PC,OBC,BP)
Shape(OAB,BO,OA)
Collinear(AOBP)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),30)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PC,O)
Goal: Value(LengthOfLine(BP))
Answer: 5
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,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,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
triangle_property_angle_sum(1,CAO)
triangle_property_angle_sum(1,COP)
adjacent_complementary_angle(1,AOC,COB)
tangent_of_circle_property_perpendicular(2,PC,O,O)
sine_theorem(1,OPC)
line_addition(1,OB,BP)
Problem: As shown in the diagram, ∠OCB=55°, O is the center of circle O, AC is the diameter of ⊙O, the tangent to circle O is PA, PB is the tangent to circle O. Find the measure of ∠BPA.
问题：如图所示，∠OCB=55°，⊙O的圆心为O，AC是⊙O的直径，PA是⊙O的切线，PB是⊙O的切线。求∠BPA的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OC,CB,BO)
Shape(OBA,AB)
Shape(OAC,CO,OA)
Shape(OCB,BC)
Shape(OBA,BP,PA)
Collinear(AOC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OCB),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 70
Theorem sequence:
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,OCB)
adjacent_complementary_angle(1,AOB,BOC)
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, ∠COA=70°, O is the center of circle O, the diameter of ⊙O is AB, the tangent to ⊙O is DA. Find the measure of ∠ADB.
问题：如图所示，∠COA=70°，O是圆O的圆心，AB是圆O的直径，DA是⊙O的切线。求∠ADB的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BC)
Shape(BO,OC,CB)
Shape(OBA,AO,OB)
Shape(AD,DC,OAC)
Collinear(BCD)
Collinear(BOA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ADB))
Answer: 55
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DA,O,O)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
triangle_property_angle_sum(1,BAD)
Problem: As shown in the diagram, ∠CDE=70°, the center of circle O is O, DC is the tangent to ⊙O, ABCD is a ▱. Find the measure of ∠CEB.
问题：如图所示，∠CDE=70°，O是⊙O的圆心，⊙O的切线为DC，BA和CD是平行四边形ABCD的一组对边。求∠CEB的大小。
Construction (CDL):
Shape(AB,BF,FA)
Shape(EA,AF,FE)
Shape(EF,FC,CE)
Shape(FB,BC,CF)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCE,EC)
Shape(OEA,AE)
Shape(CD,DE,OCE)
Collinear(AFC)
Collinear(AED)
Collinear(BFE)
Cocircular(O,ABCE)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(CEB))
Answer: 70
Theorem sequence:
circle_property_circular_power_tangent_and_segment_angle(1,DC,DEA,O)
angle_addition(1,CEF,FEA)
parallel_property_alternate_interior_angle(1,AD,BC)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,E)
arc_property_circumference_angle_external(1,OAC,E)
arc_property_circumference_angle_external(1,OCE,A)
Problem: As shown in the diagram, AB=10, AO=5, the radius of ⊙O is 3, O is the center of ⊙O, AE is the tangent to circle O, the tangent to circle O is AF, BE is the tangent to circle O, the tangent to circle O is BG, quadrilateral ABCD is a rhombus. Find the length of line BG.
问题：如图所示，AB=10，AO=5，圆O的半径为3，O是⊙O的圆心，圆O的切线为AE，圆O的切线为AF，BE是圆O的切线，圆O的切线为BG，ABCD是菱形。求直线BG的长度。
Construction (CDL):
Shape(HA,AE,OHE)
Shape(OHE,EO,OH)
Shape(FA,AH,OFH)
Shape(OFH,HO,OF)
Shape(OEG,GO,OE)
Shape(EB,BG,OEG)
Shape(OGF,FO,OG)
Shape(GB,BC,CG)
Shape(OGF,GC,CD,DF)
Collinear(AEB)
Collinear(AHO)
Collinear(AFD)
Cocircular(O,HEGF)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AO),5)
Equal(RadiusOfCircle(O),3)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AE,O)
IsTangentOfCircle(AF,O)
IsTangentOfCircle(BE,O)
IsTangentOfCircle(BG,O)
Rhombus(ABCD)
Goal: Value(LengthOfLine(BG))
Answer: 6
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AE,O,O)
radius_of_circle_property_length_equal(1,OE,O)
right_triangle_judgment_angle(1,AEO)
right_triangle_property_pythagorean(1,AEO)
line_addition(1,AE,EB)
tangent_of_circle_property_length_equal(1,BE,BG,O)
Problem: As shown in the diagram, ∠APB=50°, the center of ⊙O is O, the tangent to ⊙O is PA, the tangent to ⊙O is PB. Find the measure of ∠BCA.
问题：如图所示，∠APB=50°，圆O的圆心为O，圆O的切线为PA，圆O的切线为PB。求∠BCA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(CA,AO,OB,BC)
Shape(OAB,BO,OA)
Shape(OCA,AC)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APB),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 65
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_external(1,OAB,C)
Problem: As shown in the diagram, ∠CPA=26°, the center of ⊙O is O, the tangent to circle O is PA. Find the measure of ∠ABC.
问题：如图所示，∠CPA=26°，O是⊙O的圆心，⊙O的切线为PA。求∠ABC的大小。
Construction (CDL):
Shape(CD,DB,BC)
Shape(DA,AO,OD)
Shape(OCA,AD,DC)
Shape(OBC,CB)
Shape(CP,PA,OCA)
Shape(OAB,BD,DO,OA)
Collinear(PCDO)
Collinear(ADB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CPA),26)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 32
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
triangle_property_angle_sum(1,AOP)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠ACD=45°, the center of circle O is O, the tangent to circle O is CA. Find the measure of ∠DBA.
问题：如图所示，∠ACD=45°，⊙O的圆心为O，圆O的切线为CA。求∠DBA的大小。
Construction (CDL):
Shape(OAD,DO,OA)
Shape(DB,BO,OD)
Shape(ODB,BD)
Shape(OBA,AO,OB)
Shape(AC,CD,OAD)
Collinear(AOB)
Collinear(ODC)
Cocircular(O,BAD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),45)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CA,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 45/2
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
triangle_property_angle_sum(1,OAC)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠ECB=35°, O is the center of circle O, the diameter of circle O is AB, the tangent to circle O is EC. Find the measure of ∠CDA.
问题：如图所示，∠ECB=35°，O是⊙O的圆心，⊙O的直径为AB，圆O的切线为EC。求∠CDA的大小。
Construction (CDL):
Shape(CD,DB,BC)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(BD,DA,AB)
Shape(OAB,BO,OA)
Shape(EC,CB)
Collinear(AOB)
Cocircular(O,CDAB)
Conditions (CDL):
Equal(MeasureOfAngle(ECB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(EC,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 125
Theorem sequence:
circle_property_angle_of_osculation(2,OBC,E)
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,OBC,D)
angle_addition(1,CDB,BDA)
Problem: As shown in the diagram, ∠APB=70°, the center of ⊙O is O, PA is the tangent to ⊙O, the tangent to ⊙O is PB. Find the measure of ∠BCA.
问题：如图所示，∠APB=70°，圆O的圆心为O，PA是⊙O的切线，⊙O的切线为PB。求∠BCA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(OAB,BO,OA)
Shape(CA,AO,OB,BC)
Shape(OCA,AC)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APB),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
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_external(1,OAB,C)
Problem: As shown in the diagram, ∠CDE=30°, the radius of circle O is 2, O is the center of ⊙O, AC is the tangent to ⊙O, the tangent to circle O is BC, EF∥AB. Find the length of line EF.
问题：如图所示，∠CDE=30°，⊙O的半径为2，O是⊙O的圆心，⊙O的切线为AC，BC是圆O的切线，EF∥AB。求直线EF的长度。
Construction (CDL):
Shape(OEC,CM,MO,OE)
Shape(OEC,CD,DE)
Shape(OCF,FM,MC)
Shape(OFE,EM,MF)
Shape(AC,CM)
Shape(OE,EM,MO)
Shape(MC,CB)
Collinear(ACB)
Collinear(OMC)
Collinear(EMF)
Cocircular(O,ECFD)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),30)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AC,O)
IsTangentOfCircle(BC,O)
ParallelBetweenLine(EF,AB)
Goal: Value(LengthOfLine(EF))
Answer: 2*sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OE,O)
arc_property_circumference_angle_external(1,OEC,D)
arc_property_center_angle(1,OEC,O)
tangent_of_circle_property_perpendicular(2,AC,O,O)
parallel_property_collinear_extend(3,EF,AB,M)
parallel_property_collinear_extend(3,BA,ME,C)
parallel_property_corresponding_angle(2,CA,ME,O)
sine_theorem(1,EMO)
circle_property_chord_perpendicular_bisect_chord(1,O,OM,EF)
line_addition(1,EM,MF)
Problem: As shown in the diagram, ∠DAB=30°, O is the center of ⊙O, DC is the diameter of circle O, AB is the tangent to ⊙O. Find the measure of ∠BCO.
问题：如图所示，∠DAB=30°，O是⊙O的圆心，DC是圆O的直径，⊙O的切线为AB。求∠BCO的大小。
Construction (CDL):
Shape(ODB,BO,OD)
Shape(OBC,CB)
Shape(CO,OB,BC)
Shape(OCD,DO,OC)
Shape(DA,AB,ODB)
Collinear(ADOC)
Cocircular(O,DBC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DC,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 30
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,BOA)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠BAD=66°, O is the center of ⊙O, the tangent to circle O is DA. Find the measure of ∠ABO.
问题：如图所示，∠BAD=66°，圆O的圆心为O，DA是圆O的切线。求∠ABO的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(BO,OA,AB)
Shape(OBA,AO,OB)
Shape(OA,AD)
Shape(BA,AD)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),66)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 24
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,DA,O,O)
angle_addition(1,OAB,BAD)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
Problem: As shown in the diagram, the diameter of ⊙O is 2, ∠CAO=30°, the center of circle O is O, AB is the diameter of circle O, DC is the tangent to circle O. Find the length of line CD.
问题：如图所示，⊙O的直径为2，∠CAO=30°，⊙O的圆心为O，圆O的直径为AB，DC是圆O的切线。求直线CD的长度。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OB,BC,CO)
Shape(AO,OC,CA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(BD,DC,OBC)
Collinear(DBOA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(DiameterOfCircle(O),2)
Equal(MeasureOfAngle(CAO),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,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,OC,O)
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)
sine_theorem(1,COD)
Problem: As shown in the diagram, ∠BEC=50°, the center of ⊙O is O, AB is the diameter of circle O, the tangent to ⊙O is EC. Find the measure of ∠CDB.
问题：如图所示，∠BEC=50°，⊙O的圆心为O，AB是圆O的直径，EC是圆O的切线。求∠CDB的大小。
Construction (CDL):
Shape(OBC,CO,OB)
Shape(OBC,CD,DB)
Shape(AO,OC,OA)
Shape(OCA,AC)
Shape(OAB,BO,OA)
Shape(BE,EC,OBC)
Collinear(AOBE)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BEC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(EC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 20
Theorem sequence:
tangent_of_circle_property_perpendicular(2,EC,O,O)
triangle_property_angle_sum(1,COE)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BPA=70°, the center of ⊙O is O, the tangent to circle O is PA, PB is the tangent to ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠BPA=70°，圆O的圆心为O，⊙O的切线为PA，圆O的切线为PB。求∠ACB的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(AC,CB,BO,OA)
Shape(OBA,BP,PA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 55
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, AB=1, ∠ABO=60°, O is the center of ⊙O, BA is the tangent to ⊙O. Find the length of line OA.
问题：如图所示，AB=1，∠ABO=60°，O是圆O的圆心，BA是⊙O的切线。求直线OA的长度。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCA,AO,OC)
Shape(AB,BC,OAC)
Collinear(OCB)
Cocircular(O,AC)
Conditions (CDL):
Equal(LengthOfLine(AB),1)
Equal(MeasureOfAngle(ABO),60)
IsCentreOfCircle(O,O)
IsTangentOfCircle(BA,O)
Goal: Value(LengthOfLine(OA))
Answer: sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(1,BA,O,O)
triangle_property_angle_sum(1,OAB)
sine_theorem(1,ABO)
Problem: As shown in the diagram, ∠BCD=132°, O is the center of circle O, BE is the diameter of circle O, AB is the tangent to ⊙O, AD is the tangent to circle O. Find the measure of ∠BAD.
问题：如图所示，∠BCD=132°，O是圆O的圆心，圆O的直径为BE，AB是圆O的切线，⊙O的切线为AD。求∠BAD的大小。
Construction (CDL):
Shape(BA,AD,OBC,OCD)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(BC,CD,DO,OB)
Shape(OD,DE,EO)
Shape(ODE,ED)
Shape(BA,AD,DO,OB)
Shape(OEB,BO,OE)
Collinear(BOE)
Cocircular(O,BCDE)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),132)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
IsTangentOfCircle(AB,O)
IsTangentOfCircle(AD,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 84
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
tangent_of_circle_property_perpendicular(1,AB,O,O)
tangent_of_circle_property_perpendicular(2,AD,O,O)
quadrilateral_property_angle_sum(1,BADO)
Problem: As shown in the diagram, ∠OAC=20°, O is the center of ⊙O, DC is the tangent to ⊙O. Find the measure of ∠CDB.
问题：如图所示，∠OAC=20°，圆O的圆心为O，DC是⊙O的切线。求∠CDB的大小。
Construction (CDL):
Shape(CD,DB,OCB)
Shape(OCB,BO,OC)
Shape(AC,CO,OA)
Shape(OAC,CA)
Shape(OBA,AO,OB)
Collinear(AOBD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OAC),20)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
tangent_of_circle_property_perpendicular(1,DC,O,O)
triangle_property_angle_sum(1,CDO)
Problem: As shown in the diagram, ∠DOB=100°, O is the center of circle O, the diameter of circle O is CB, the tangent to ⊙O is AB. Find the measure of ∠BAC.
问题：如图所示，∠DOB=100°，O是圆O的圆心，圆O的直径为CB，圆O的切线为AB。求∠BAC的大小。
Construction (CDL):
Shape(OBD,DO,OB)
Shape(ODC,CD)
Shape(OD,DC,CO)
Shape(OCB,BO,OC)
Shape(BA,AD,OBD)
Collinear(CDA)
Collinear(COB)
Cocircular(O,CBD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),100)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CB,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,C)
tangent_of_circle_property_perpendicular(1,AB,O,O)
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, ∠ACB=115°, the center of circle O is O, AB is the tangent to ⊙O. Find the measure of ∠BAC.
问题：如图所示，∠ACB=115°，O是圆O的圆心，AB是圆O的切线。求∠BAC的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OB,BC,CO)
Shape(OCB,BO,OC)
Shape(BA,AC,OBC)
Collinear(OCA)
Cocircular(O,BC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),115)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(1,AB,O,O)
adjacent_complementary_angle(1,ACB,BCO)
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)
triangle_property_angle_sum(1,OBA)
Problem: As shown in the diagram, ∠BPA=60°, the center of ⊙O is O, the tangent to ⊙O is PA, PB is the tangent to circle O. Find the measure of ∠BAO.
问题：如图所示，∠BPA=60°，O是⊙O的圆心，PA是圆O的切线，PB是⊙O的切线。求∠BAO的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(AO,OB,BA)
Shape(OAB,BO,OA)
Shape(PA,OBA,BP)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),60)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BAO))
Answer: 30
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)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,AOB)
Problem: As shown in the diagram, the radius of ⊙O is 5, O is the center of circle O, the tangent to ⊙O is PA. Find the length of line OA.
问题：如图所示，⊙O的半径为5，圆O的圆心为O，⊙O的切线为PA。求直线OA的长度。
Construction (CDL):
Shape(OA,AP)
Cocircular(O,A)
Conditions (CDL):
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(OA))
Answer: 5
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)