SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCE=52°, the center of ⊙O is O, AC is the diameter of ⊙O. Find the measure of ∠CDE.
问题：如图所示，∠BCE=52°，O是圆O的圆心，⊙O的直径为AC。求∠CDE的大小。
Construction (CDL):
Shape(AB,BE,EO,OA)
Shape(DE,EC,CD)
Shape(EB,BC,CE)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AO,OE,ED,ODA)
Collinear(AOEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCE),52)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(CDE))
Answer: 38
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,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, AD=BD, ∠ACD=90°, ∠BAC=67.5°. Find the value of sin(CD).
问题：如图所示，AD=BD，∠ACD=90°，∠BAC=67.5°。求sin(CD)的值。
Construction (CDL):
Shape(AC,CD,DA)
Shape(AD,DB,BA)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(BD))
Equal(MeasureOfAngle(ACD),90)
Equal(MeasureOfAngle(BAC),67.5)
Goal: Value(Sin(MeasureOfAngle(CDA)))
Answer: sqrt(2)/2
Theorem sequence:
triangle_property_angle_sum(1,BAC)
isosceles_triangle_judgment_line_equal(1,DBA)
isosceles_triangle_property_angle_equal(1,DBA)
triangle_property_angle_sum(1,ADB)
adjacent_complementary_angle(1,CDA,ADB)
Problem: As shown in the diagram, AB=2, BC=1, ∠ACB=90°. Find the value of sin(CB).
问题：如图所示，AB=2，BC=1，∠ACB=90°。求sin(CB)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BC),1)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Sin(MeasureOfAngle(CBA)))
Answer: sqrt(3)/2
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
cosine_theorem(1,BAC)
Problem: As shown in the diagram, ∠OBC=50°, the center of ⊙O is O, AB is the diameter of ⊙O, D is the midpoint of ⌒OCA. Find the measure of ∠BCD.
问题：如图所示，∠OBC=50°，圆O的圆心为O，圆O的直径为AB，D平分弧OCA。求∠BCD的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(DO,OC,CD)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(D,OCA)
Goal: Value(MeasureOfAngle(BCD))
Answer: 115
Theorem sequence:
angle_addition(1,BCO,OCD)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
adjacent_complementary_angle(1,AOC,COB)
congruent_arc_judgment_length_equal(1,OCD,ODA)
congruent_arc_property_measure_equal(1,OCD,ODA)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,ODA,O)
angle_addition(1,AOD,DOC)
isosceles_triangle_judgment_line_equal(1,OCD)
isosceles_triangle_property_angle_equal(1,OCD)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠BAC=30°. Find the value of tan(BA).
问题：如图所示，∠BAC=30°。求tan(BA)的值。
Construction (CDL):
Shape(CB,BA,AC)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
Goal: Value(Tan(MeasureOfAngle(BAC)))
Answer: sqrt(3)/3
Theorem sequence:
Problem: As shown in the diagram, ∠EAB=70°, ∠ECD=50°, the center of circle O is O. Find the value of sin(BE).
问题：如图所示，∠EAB=70°，∠ECD=50°，O是⊙O的圆心。求sin(BE)的值。
Construction (CDL):
Shape(AB,BE,EA)
Shape(DE,EC,CD)
Shape(BA.OAB)
Shape(AE,ED,ODA)
Shape(DC,OCD)
Shape(CE,EB,OBC)
Collinear(BED)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),70)
Equal(MeasureOfAngle(ECD),50)
IsCentreOfCircle(O,O)
Goal: Value(Sin(MeasureOfAngle(BEA)))
Answer: sqrt(3)/2
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, Add(MeasureOfAngle(OAB)=MeasureOfAngle(BCO)), the center of ⊙O is O. Find the measure of ∠AOC.
问题：如图所示，Add(MeasureOfAngle(OAB)=MeasureOfAngle(BCO))，O是⊙O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BC,CO)
Shape(AO,OC,OCA)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(OAB),MeasureOfAngle(BCO)),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 150
Theorem sequence:
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)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OBC)
angle_addition(1,ABO,OBC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠BCA=90°, ∠CAB=30°. Find sin(CAB)+cos(ABC).
问题：如图所示，∠BCA=90°，∠CAB=30°。求sin(CAB)+cos(ABC)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CAB),30)
Goal: Value(Add(Sin(MeasureOfAngle(CAB)),Cos(MeasureOfAngle(ABC))))
Answer: 1
Theorem sequence:
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠ABO=25°, ∠OCA=30°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠ABO=25°，∠OCA=30°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(BA,OAB)
Shape(CO,OB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),25)
Equal(MeasureOfAngle(OCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 110
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
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)
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,AOC)
angle_addition(1,BOA,AOC)
round_angle(1,COB,BOC)
Problem: As shown in the diagram, ∠OBC=65°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠OCA.
问题：如图所示，∠OBC=65°，圆O的圆心为O，圆O的直径为AB。求∠OCA的大小。
Construction (CDL):
Shape(AO,OC,CA)
Shape(CO,OB,BC)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),65)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 25
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
adjacent_complementary_angle(1,AOC,COB)
triangle_property_angle_sum(1,OCA)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, AB=2, AE=4, BC=1, DE=2, ∠ACB=90°, ∠CDE=90°. Find the value of sin(BA).
问题：如图所示，AB=2，AE=4，BC=1，DE=2，∠ACB=90°，∠CDE=90°。求sin(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Shape(BC,CD,DE,EB)
Collinear(ABE)
Collinear(ACD)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(AE),4)
Equal(LengthOfLine(BC),1)
Equal(LengthOfLine(DE),2)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(CDE),90)
Goal: Value(Sin(MeasureOfAngle(BAC)))
Answer: 1/2
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
cosine_theorem(1,ACB)
Problem: As shown in the diagram, AB=AC, ∠BCE=65°, O is the center of circle O, BA is parallel to CD. Find the measure of ∠EBC.
问题：如图所示，AB=AC，∠BCE=65°，圆O的圆心为O，BA∥CD。求∠EBC的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(DE,EC,CD)
Shape(EB,BC,CE)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AE,ED,ODA)
Collinear(AEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(BCE),65)
IsCentreOfCircle(O,O)
ParallelBetweenLine(BA,CD)
Goal: Value(MeasureOfAngle(EBC))
Answer: 15
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
parallel_property_alternate_interior_angle(1,BA,CD)
angle_addition(1,ABE,EBC)
Problem: As shown in the diagram, BC=4, CA=3, ∠BCA=90°. Find the value of tan(AB).
问题：如图所示，BC=4，CA=3，∠BCA=90°。求tan(AB)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(CA),3)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Tan(MeasureOfAngle(ABC)))
Answer: 3/4
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
cosine_theorem(1,BCA)
Problem: As shown in the diagram, CD=8, ∠OAC=22.5°, the center of circle O is O, the diameter of ⊙O is AB, CE is perpendicular to OE. Find the length of line OC.
问题：如图所示，CD=8，∠OAC=22.5°，⊙O的圆心为O，AB是圆O的直径，CE垂直于OE。求直线OC的长度。
Construction (CDL):
Shape(AC,CO,OA)
Shape(OC,CE,EO)
Shape(OE,ED,DO)
Shape(CA,OAC)
Shape(AO,OD,ODA)
Shape(BE,EC,OCB)
Shape(DE,EB,OBD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(CD),8)
Equal(MeasureOfAngle(OAC),22.5)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(OC))
Answer: 4*sqrt(2)
Theorem sequence:
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,OD,O)
adjacent_complementary_angle(1,CEO,OED)
isosceles_triangle_judgment_line_equal(1,OCD)
isosceles_triangle_property_angle_equal(1,OCD)
mirror_congruent_triangle_judgment_aas(1,OCE,OED)
mirror_congruent_triangle_property_line_equal(1,OCE,OED)
line_addition(1,CE,ED)
triangle_property_angle_sum(1,ACE)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
angle_addition(1,ACO,OCE)
triangle_property_angle_sum(1,OCE)
isosceles_triangle_judgment_angle_equal(1,EOC)
right_triangle_judgment_angle(1,CEO)
right_triangle_property_pythagorean(1,CEO)
Problem: As shown in the diagram, ∠ABE=24°, ∠EAB=60°, the center of circle O is O. Find the measure of ∠ECD.
问题：如图所示，∠ABE=24°，∠EAB=60°，O是圆O的圆心。求∠ECD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(DE,EC,CD)
Shape(AE,ED,ODA)
Shape(BA,OAB)
Shape(DC,OCD)
Shape(CE,EB,OBC)
Collinear(AEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),24)
Equal(MeasureOfAngle(EAB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ECD))
Answer: 24
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠BOA=110°, the center of ⊙O is O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=110°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(OA,AC,CB,BO)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 125
Theorem sequence:
round_angle(1,AOB,BOA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠COB=40°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠CDA.
问题：如图所示，∠COB=40°，圆O的圆心为O，圆O的直径为AB。求∠CDA的大小。
Construction (CDL):
Shape(DA,AO,OC,CD)
Shape(CO,OB,BC)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 110
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
round_angle(1,AOC,COA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠OCA=30°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠OCA=30°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OC,CA,AO)
Shape(OCA,AC)
Shape(OAB,BO,OA)
Shape(OBC,CO,OB)
Collinear(COB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, AB=AC, ∠CAB=36°. Find the measure of ∠BCA.
问题：如图所示，AB=AC，∠CAB=36°。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),36)
Goal: Value(MeasureOfAngle(BCA))
Answer: 72
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠DOB=50°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠DOB=50°，O是⊙O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(OBD,DO,OB)
Shape(ODA,AD)
Shape(OAB,BO)
Shape(DA,AB,BO,OD)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠BCA=46°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=46°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BO,OA)
Shape(OAB,BC,CA)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),46)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 92
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOC=50°, the center of circle O is O, CE is perpendicular to OE. Find the measure of ∠DAB.
问题：如图所示，∠BOC=50°，O是圆O的圆心，CE⊥OE。求∠DAB的大小。
Construction (CDL):
Shape(OCB,BE,EC)
Shape(OC,CE,EO)
Shape(OBD,DA,AB)
Shape(OCB,OBD,DC)
Shape(ODC,CE,ED)
Shape(OBD,DE,EB)
Collinear(CED)
Collinear(OEB)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(MeasureOfAngle(DAB))
Answer: 25
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OCD,OEB)
arc_property_center_angle(1,OCB,O)
congruent_arc_judgment_length_equal(1,OCB,OBD)
congruent_arc_property_measure_equal(1,OCB,OBD)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠ADC=35°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=35°，⊙O的圆心为O，圆O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OAC,AD,DC)
Shape(OAB,BO,OA)
Shape(OBC,CB)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠BCD=130°. Find the measure of arc OBD.
问题：如图所示，∠BCD=130°。求⌒OBD的角度。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(OB,BC,CD,DO)
Shape(AB,BO,OD,DA)
Shape(ODA,AD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),130)
Goal: Value(MeasureOfArc(OBD))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠CAO=32°, O is the center of ⊙O, CB is the diameter of circle O. Find the measure of ∠ABO.
问题：如图所示，∠CAO=32°，O是圆O的圆心，圆O的直径为CB。求∠ABO的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(OBC,CO,OB)
Shape(OC,CA,AO)
Shape(OA,AB,BO)
Collinear(COB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CB,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 58
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAB,O)
angle_addition(1,CAO,OAB)
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)
Problem: As shown in the diagram, ∠BCA=35°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OAB,BC,CA)
Shape(OBA,AO,OB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 70
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OAB=25°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠OAB=25°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(PA,AO,OB,BP)
Shape(OA,AB,BO)
Shape(AC,CB,BA)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OBP,PB)
Shape(OPA,AP)
Cocircular(O,ACBP)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 115
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)
arc_property_center_angle(1,OAB,O)
triangle_property_angle_sum(1,OAB)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, ∠BAD=25°, O is the center of ⊙O, AE is perpendicular to CE, DE is perpendicular to EE. Find the measure of ∠DGA.
问题：如图所示，∠BAD=25°，O是⊙O的圆心，AE垂直于CE，DE垂直于EE。求∠DGA的大小。
Construction (CDL):
Shape(FG,OCG,CF)
Shape(OCG,GC)
Shape(GA,AO,OE,EC,CG)
Shape(OBC,CE,EB)
Shape(EO,OA,AD,DE)
Shape(ODB,BD)
Shape(BE,ED,DB)
Shape(GA,AD,DG)
Shape(OAD,DA)
Shape(OGA,AG)
Collinear(AOEB)
Collinear(FCED)
Collinear(AGF)
Cocircular(O,ADBCG)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,CE)
PerpendicularBetweenLine(DE,EA)
Goal: Value(MeasureOfAngle(DGA))
Answer: 65
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,G)
Problem: As shown in the diagram, ∠BAC=20°, the center of ⊙O is O. Find the measure of ∠ADC.
问题：如图所示，∠BAC=20°，圆O的圆心为O。求∠ADC的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(CB,BO,OA,AC)
Shape(AD,DC,CA)
Collinear(AOB)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 110
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
triangle_property_angle_sum(1,CBA)
diameter_of_circle_property_right_angle(1,ACB,O)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_internal(1,OAC,D)
Problem: As shown in the diagram, ∠COA=120°, O is the center of ⊙O. Find the measure of ∠CBA.
问题：如图所示，∠COA=120°，O是⊙O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(AO,OC,CB,BA)
Shape(OAC,CO,OA)
Shape(OCB,BC)
Shape(OBA,AB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠ABC=29°, ∠ACD=99°, AD is the angle bisector of ∠EAC. Find the measure of ∠EAD.
问题：如图所示，∠ABC=29°，∠ACD=99°，AD是∠EAC的角平分线。求∠EAD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(EA,AD)
Collinear(BAE)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),29)
Equal(MeasureOfAngle(ACD),99)
IsBisectorOfAngle(AD,EAC)
Goal: Value(MeasureOfAngle(EAD))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,ACD)
triangle_property_angle_sum(1,ABD)
adjacent_complementary_angle(1,EAD,DAB)
Problem: As shown in the diagram, ∠CAP=35°, ∠CPB=78°, O is the center of ⊙O. Find the measure of ∠DBP.
问题：如图所示，∠CAP=35°，∠CPB=78°，圆O的圆心为O。求∠DBP的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CP,PB)
Shape(OAD,DP,PA)
Shape(ODB,BD)
Shape(PC,CA,AP)
Shape(BP,PD,DB)
Collinear(CPD)
Collinear(APB)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAP),35)
Equal(MeasureOfAngle(CPB),78)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBP))
Answer: 43
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,BPD)
adjacent_complementary_angle(1,CPB,BPD)
Problem: As shown in the diagram, ∠OCB=21°, the center of ⊙O is O, CO is parallel to AB. Find the measure of ∠AOC.
问题：如图所示，∠OCB=21°，O是圆O的圆心，CO∥AB。求∠AOC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OA,AB)
Shape(OC,CB)
Shape(OBC,CB)
Shape(OCA,AO,OC)
Shape(OCA,AB,BC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCB),21)
IsCentreOfCircle(O,O)
ParallelBetweenLine(CO,AB)
Goal: Value(MeasureOfAngle(AOC))
Answer: 42
Theorem sequence:
parallel_property_alternate_interior_angle(1,CO,AB)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠BOC=40°, BO bisects ∠CBD, CO bisects ∠ECB. Find the measure of ∠CAB.
问题：如图所示，∠BOC=40°，BO平分∠CBD，CO是∠ECB的角平分线。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BO,OC,CB)
Shape(OB,BD)
Shape(EC,CO)
Collinear(ABD)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),40)
IsBisectorOfAngle(BO,CBD)
IsBisectorOfAngle(CO,ECB)
Goal: Value(MeasureOfAngle(CAB))
Answer: 100
Theorem sequence:
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BOC)
angle_addition(1,CBO,OBD)
angle_addition(1,ECO,OCB)
adjacent_complementary_angle(1,ABC,CBD)
adjacent_complementary_angle(1,ECB,BCA)
Problem: As shown in the diagram, ∠OAB=55°, the center of circle O is O. Find the measure of ∠BDA.
问题：如图所示，∠OAB=55°，O是圆O的圆心。求∠BDA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(AB,BD,DA)
Shape(AB,BC,CO,OA)
Shape(OCD,DB,BC)
Shape(ODA,AD)
Collinear(AOC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 35
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOC,O)
diameter_of_circle_property_right_angle(1,ABC,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠ODE=50°, the center of circle O is O, AO∥DE. Find the measure of ∠OCA.
问题：如图所示，∠ODE=50°，O是圆O的圆心，AO平行于DE。求∠OCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAD,DO,OA)
Shape(ODE,ED)
Shape(OEC,CO,OD,DE)
Shape(OC,CA,AO)
Collinear(COD)
Cocircular(O,CADE)
Conditions (CDL):
Equal(MeasureOfAngle(ODE),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(OCA))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠CAB=60°, the radius of circle O is 2, the center of circle O is O, BD is perpendicular to OD. Find the length of line BC.
问题：如图所示，∠CAB=60°，圆O的半径为2，O是⊙O的圆心，BD垂直于OD。求直线BC的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CD,DB)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(AB,BD,DC,CA)
Shape(OCA,AC)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(BC))
Answer: 2*sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
arc_property_circumference_angle_external(1,OBC,A)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BC)
angle_addition(1,COD,DOB)
perpendicular_bisector_property_bisector(1,OD,BC)
right_triangle_judgment_angle(1,BDO)
arc_property_center_angle(1,OBC,O)
sine_theorem(1,BDO)
sine_theorem(1,ODC)
sine_theorem(1,BCO)
right_triangle_property_pythagorean(1,BDO)
Problem: As shown in the diagram, ∠COA=140°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠COA=140°，O是⊙O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AO,OC)
Shape(OA,AB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)