SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AD=AC, ∠BAD=30°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠EOC.
问题：如图所示，AD=AC，∠BAD=30°，⊙O的圆心为O，圆O的直径为AB。求∠EOC的大小。
Construction (CDL):
Shape(ODB,BC,CD)
Shape(OAD,DA)
Shape(DC,CO,OA,AD)
Shape(OBE,EC,CB)
Shape(CE,EO,OC)
Shape(OE,EA,AO)
Shape(OEA,AE)
Collinear(BCOA)
Collinear(ECD)
Cocircular(O,BEAD)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(AC))
Equal(MeasureOfAngle(BAD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(EOC))
Answer: 30
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ADC)
isosceles_triangle_property_angle_equal(1,ADC)
triangle_property_angle_sum(1,ADC)
arc_property_circumference_angle_external(1,OEA,D)
arc_property_center_angle(1,OEA,O)
flat_angle(1,AOC)
angle_addition(1,AOE,EOC)
Problem: As shown in the diagram, ∠CDA=115°, O is the center of circle O, the diameter of ⊙O is AB, CD is the tangent to ⊙O. Find the measure of ∠ACD.
问题：如图所示，∠CDA=115°，⊙O的圆心为O，圆O的直径为AB，CD是⊙O的切线。求∠ACD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(DO,OB,OBD)
Shape(OBD,BC,CD)
Shape(OA,OAB,BO)
Collinear(AOBC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),115)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(2,CD,O,O)
angle_addition(1,CDO,ODA)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,DAC)
Problem: As shown in the diagram, ∠BTA=40°, O is the center of ⊙O, AB is the diameter of circle O, TA is the tangent to ⊙O. Find the measure of ∠CDB.
问题：如图所示，∠BTA=40°，⊙O的圆心为O，⊙O的直径为AB，TA是⊙O的切线。求∠CDB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(BC,CE,EB)
Shape(ODB,BD)
Shape(BE,ED,DB)
Shape(EC,CA,AO,OE)
Shape(DE,EO,OA,OAD)
Shape(OCA,AC)
Shape(CT,TA,OCA)
Collinear(CED)
Collinear(BEOA)
Collinear(TCB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BTA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(TA,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(2,TA,O,O)
triangle_property_angle_sum(1,BTA)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠FOA=40°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠BEF.
问题：如图所示，∠FOA=40°，⊙O的圆心为O，圆O的直径为AB。求∠BEF的大小。
Construction (CDL):
Shape(OEA,AC,CE)
Shape(OBE,EB)
Shape(EC,CO,OB,BE)
Shape(OAF,FC,CA)
Shape(CF,FO,OC)
Shape(OFB,BO,OF)
Collinear(ACOB)
Collinear(FCE)
Cocircular(O,AFBE)
Conditions (CDL):
Equal(MeasureOfAngle(FOA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BEF))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,BOF,FOA)
arc_property_center_angle(1,OFB,O)
arc_property_circumference_angle_external(1,OFB,E)
Problem: As shown in the diagram, ∠CBO=70°, O is the center of ⊙O, AB is the diameter of circle O, PA is the tangent to circle O, PO∥CB. Find the measure of ∠OPC.
问题：如图所示，∠CBO=70°，O是⊙O的圆心，AB是⊙O的直径，PA是圆O的切线，PO平行于CB。求∠OPC的大小。
Construction (CDL):
Shape(AP,PD,OAD)
Shape(OAD,DO,OA)
Shape(DP,PC,ODC)
Shape(ODC,CO,OD)
Shape(OC,CB,BO)
Shape(OCB,BC)
Shape(OB,OBA,AO)
Collinear(AOB)
Collinear(PDO)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PA,O)
ParallelBetweenLine(PO,CB)
Goal: Value(MeasureOfAngle(OPC))
Answer: 20
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
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)
parallel_property_corresponding_angle(2,BC,OP,A)
parallel_property_alternate_interior_angle(2,PO,CB)
triangle_property_angle_sum(1,APO)
radius_of_circle_property_length_equal(1,OA,O)
mirror_congruent_triangle_judgment_sas(1,OPC,OAP)
mirror_congruent_triangle_property_angle_equal(1,PCO,POA)
Problem: As shown in the diagram, AB=AC, ∠BCA=70°, the center of ⊙O is O, O is the center of the circumscribed circle of triangle ABC. Find the measure of ∠ABD.
问题：如图所示，AB=AC，∠BCA=70°，⊙O的圆心为O，三角形ABC的外心为O。求∠ABD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(OB,BE,EO)
Shape(DO,OC,CD)
Shape(OE,EC,CO)
Shape(OBC,CB)
Shape(OCA,AD,DC)
Collinear(ADC)
Collinear(BOD)
Collinear(AOE)
Collinear(BEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(BCA),70)
IsCentreOfCircle(O,O)
IsCircumcenterOfTriangle(O,ABC)
Goal: Value(MeasureOfAngle(ABD))
Answer: 20
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
mirror_congruent_triangle_judgment_sss(1,OAB,OCA)
mirror_congruent_triangle_property_angle_equal(1,ABO,AOC)
angle_addition(1,CAO,OAB)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)