SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ACB=60°, ∠HBC=25°, CF is parallel to HB, GE∥CF. Find the measure of ∠AGE.
问题：如图所示，∠ACB=60°，∠HBC=25°，CF平行于HB，GE∥CF。求∠AGE的大小。
Construction (CDL):
Shape(AG,GE,EA)
Shape(EG,GC,CF,FE)
Shape(FC,CB,BF)
Shape(HB,BC)
Collinear(AEFB)
Collinear(AGC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),60)
Equal(MeasureOfAngle(HBC),25)
ParallelBetweenLine(CF,HB)
ParallelBetweenLine(GE,CF)
Goal: Value(MeasureOfAngle(AGE))
Answer: 35
Theorem sequence:
parallel_property_alternate_interior_angle(1,CF,HB)
parallel_property_corresponding_angle(1,GE,CF,A)
angle_addition(1,GCF,FCB)
Problem: As shown in the diagram, ∠EMB=70°, MB is parallel to ND. Find the measure of ∠MND.
问题：如图所示，∠EMB=70°，MB∥ND。求∠MND的大小。
Construction (CDL):
Shape(EM,MB)
Shape(BM,MN)
Shape(MN,ND)
Collinear(EMN)
Conditions (CDL):
Equal(MeasureOfAngle(EMB),70)
ParallelBetweenLine(MB,ND)
Goal: Value(MeasureOfAngle(MND))
Answer: 70
Theorem sequence:
parallel_property_corresponding_angle(1,MB,ND,E)
Problem: As shown in the diagram, ∠AOE=52°, EO is perpendicular to DO. Find the measure of ∠DOB.
问题：如图所示，∠AOE=52°，EO⊥DO。求∠DOB的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OE)
Shape(EO,OD)
Shape(DO,OB)
Shape(BO,OC)
Collinear(AOB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(AOE),52)
PerpendicularBetweenLine(EO,DO)
Goal: Value(MeasureOfAngle(DOB))
Answer: 38
Theorem sequence:
flat_angle(1,AOB)
angle_addition(1,AOE,EOB)
angle_addition(1,EOD,DOB)
Problem: As shown in the diagram, ∠BCA=20°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=20°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OA,AF,FO)
Shape(FA,OAB,BF)
Shape(FB,BC,CF)
Shape(OBC,CB)
Shape(OF,FC,OCA,AO)
Collinear(AFC)
Collinear(OFB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=100°, the center of circle O is O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=100°，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OCB,BC)
Shape(AC,CB,BO,OA)
Shape(OB,BE,EA,AO)
Shape(OBE,EB)
Shape(OEA,AE)
Cocircular(O,ACBE)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, AC=2, BA=4, CB=3, DA=AC, ∠ACB=∠EDA, ∠BAC=∠DAE. Find the length of line ED.
问题：如图所示，AC=2，BA=4，CB=3，DA=AC，∠ACB=∠EDA，∠BAC=∠DAE。求直线ED的长度。
Construction (CDL):
Shape(BA,AC,CB)
Shape(ED,DA,AE)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(BA),4)
Equal(LengthOfLine(CB),3)
Equal(LengthOfLine(DA),LengthOfLine(AC))
Equal(MeasureOfAngle(ACB),MeasureOfAngle(EDA))
Equal(MeasureOfAngle(BAC),MeasureOfAngle(DAE))
Goal: Value(LengthOfLine(ED))
Answer: 3
Theorem sequence:
mirror_congruent_triangle_judgment_aas(2,BAC,EDA)
mirror_congruent_triangle_property_line_equal(1,ACB,AED)
Problem: As shown in the diagram, ∠AOD=150°, AO and OB are the legs of the right △ AOB, CO and OD are the legs of the right △ COD. Find the measure of ∠COB.
问题：如图所示，∠AOD=150°，三角形AOB为直角三角形，CO和OD是直角三角形COD的两个直角边。求∠COB的大小。
Construction (CDL):
Shape(AO,OE,EA)
Shape(CE,EF,FC)
Shape(FG,GB,BF)
Shape(EO,OG,GF,FE)
Shape(GO,OD,DG)
Collinear(BFEA)
Collinear(CEO)
Collinear(DGFC)
Collinear(OGB)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),150)
RightTriangle(AOB)
RightTriangle(COD)
Goal: Value(MeasureOfAngle(COB))
Answer: 30
Theorem sequence:
angle_addition(1,AOE,EOG)
angle_addition(1,AOE,EOD)
Problem: As shown in the diagram, ∠ABD=62°, ∠ABD=∠CGH, ∠ABD=∠FGB. Find the measure of ∠BCE.
问题：如图所示，∠ABD=62°，∠ABD=∠CGH，∠ABD=∠FGB。求∠BCE的大小。
Construction (CDL):
Shape(BG,GC,CB)
Shape(BC,CE,ED,DB)
Shape(FG,GB)
Shape(CG,GH)
Shape(AB,BD)
Shape(EC,CI)
Collinear(ABCI)
Collinear(DBG)
Collinear(GCE)
Collinear(FGH)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),62)
Equal(MeasureOfAngle(ABD),MeasureOfAngle(CGH))
Equal(MeasureOfAngle(ABD),MeasureOfAngle(FGB))
Goal: Value(MeasureOfAngle(BCE))
Answer: 118
Theorem sequence:
parallel_judgment_corresponding_angle(2,GF,BA,D)
parallel_property_collinear_extend(1,GF,BA,H)
parallel_property_collinear_extend(2,AB,GH,C)
parallel_property_alternate_interior_angle(2,BC,GH)
adjacent_complementary_angle(1,GCB,BCE)
Problem: As shown in the diagram, AD=1, DB=2, △ADE is similar to △ABC.. Find the ratio of similar triangles ADE and ABC.
问题：如图所示，AD=1，DB=2，△ADE相似于△ABC。求△ADE和△ABC的相似比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(CEA)
Conditions (CDL):
Equal(LengthOfLine(AD),1)
Equal(LengthOfLine(DB),2)
SimilarBetweenTriangle(ADE,ABC)
Goal: Value(RatioOfSimilarTriangle(ADE,ABC))
Answer: 1/3
Theorem sequence:
line_addition(1,AD,DB)
similar_triangle_property_line_ratio(1,EAD,CAB)
Problem: As shown in the diagram, ∠ABF=45°, ∠DEF=21°, CD is parallel to AB. Find the measure of ∠FDE.
问题：如图所示，∠ABF=45°，∠DEF=21°，CD∥AB。求∠FDE的大小。
Construction (CDL):
Shape(CF,FE)
Shape(AB,BF)
Shape(EF,FD,DE)
Collinear(CFD)
Collinear(BFE)
Conditions (CDL):
Equal(MeasureOfAngle(ABF),45)
Equal(MeasureOfAngle(DEF),21)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(FDE))
Answer: 24
Theorem sequence:
parallel_property_collinear_extend(3,CD,AB,F)
parallel_property_corresponding_angle(2,BA,FC,E)
adjacent_complementary_angle(1,CFE,EFD)
triangle_property_angle_sum(1,EFD)
Problem: As shown in the diagram, ∠ABC=38°, ∠CAB=62°. Find the measure of ∠ACD.
问题：如图所示，∠ABC=38°，∠CAB=62°。求∠ACD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),38)
Equal(MeasureOfAngle(CAB),62)
Goal: Value(MeasureOfAngle(ACD))
Answer: 100
Theorem sequence:
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,BCA,ACD)
Problem: As shown in the diagram, PQ=BP, ∠BPQ=40°. Find the measure of ∠PQB.
问题：如图所示，PQ=BP，∠BPQ=40°。求∠PQB的大小。
Construction (CDL):
Shape(O,O)
Shape(AO,OC,OCB,BQ,QA)
Shape(PQ,QB,BP)
Shape(AQ,QP,OPA)
Shape(BP,0BP)
Collinear(AOC)
Collinear(AQB)
Cocircular(O,ABP)
Conditions (CDL):
Equal(LengthOfLine(PQ),LengthOfLine(BP))
Equal(MeasureOfAngle(BPQ),40)
Goal: Value(MeasureOfAngle(PQB))
Answer: 70
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,PQB)
triangle_property_angle_sum(1,PQB)
isosceles_triangle_property_angle_equal(1,PQB)
Problem: As shown in the diagram, the diameter of circle O is equal to the length of line AB, BC=6, CA=8, O is the center of ⊙O, triangle BCA is a right triangle. Find the radius of ⊙O.
问题：如图所示，圆O的直径与直线AB的长度相等，BC=6，CA=8，O是⊙O的圆心，△BCA为直角△。求圆O的半径。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AB,BC,CA)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(DiameterOfCircle(O),LengthOfLine(AB))
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(CA),8)
IsCentreOfCircle(O,O)
RightTriangle(BCA)
Goal: Value(RadiusOfCircle(O))
Answer: 5
Theorem sequence:
right_triangle_property_pythagorean(1,BCA)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, ∠BCD=40°, the center of circle O is O. Find the measure of ∠CAD.
问题：如图所示，∠BCD=40°，圆O的圆心为O。求∠CAD的大小。
Construction (CDL):
Shape(BO,OD,DA,OAB)
Shape(OB,BD,DO)
Shape(DB,BC,CD)
Shape(DC,CA,AD)
Shape(AC,OCA)
Shape(BC,OBC)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAD))
Answer: 50
Theorem sequence:
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, the diameter of circle O is equal to the length of line DA, ∠EBA=30°, O is the center of circle O, AD is the diameter of circle O. Find the measure of ∠EAC.
问题：如图所示，⊙O的直径与直线DA的长度相等，∠EBA=30°，O是⊙O的圆心，AD是⊙O的直径。求∠EAC的大小。
Construction (CDL):
Shape(BE,EO,OD,ODB)
Shape(DO,OE,EC,CD)
Shape(DC,OCD)
Shape(EA,AC,CE)
Shape(CA,OAC)
Shape(EB,BA,AE)
Shape(BA,OBA)
Collinear(DOEA)
Collinear(BEC)
Cocircular(O,DBAC)
Conditions (CDL):
Equal(DiameterOfCircle(O),LengthOfLine(DA))
Equal(MeasureOfAngle(EBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(EAC))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
diameter_of_circle_property_right_angle(1,ACD,O)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, the diameter of ⊙O is equal to the length of line BD, ∠EAB=60°, the radius of ⊙O is 1, the center of ⊙O is O, the diameter of ⊙O is BD. Find the length of line BC.
问题：如图所示，⊙O的直径与直线BD的长度相等，∠EAB=60°，⊙O的半径为1，O是圆O的圆心，BD是圆O的直径。求直线BC的长度。
Construction (CDL):
Shape(AB,OAB)
Shape(AB,BO,OE,EA)
Shape(AE,ED,ODA)
Shape(EC,CD,DE)
Shape(CD,OCD)
Shape(EO,OB,BC,CE)
Shape(BC,OBC)
Collinear(CEA)
Collinear(BOED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(DiameterOfCircle(O),LengthOfLine(BD))
Equal(MeasureOfAngle(EAB),60)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(LengthOfLine(BC))
Answer: sqrt(3)
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
diameter_of_circle_property_right_angle(1,BCD,O)
sine_theorem(1,BCD)
Problem: As shown in the diagram, the radius of ⊙O is 2, △ABC is an equilateral △, the center of circle O is O, D is the midpoint of segment AB, E bisects segment CA. Find the length of line DE.
问题：如图所示，圆O的半径为2，△ABC为等边△，⊙O的圆心为O，D平分线段AB，E是线段CA的中点。求直线DE的长度。
Construction (CDL):
Shape(AD,DB,OAB)
Shape(AD,DE,EA)
Shape(DB,BO,OC,CE,ED)
Shape(OB,BC,CO)
Shape(BC,OBC)
Shape(CE,EA,OCA)
Collinear(ADB)
Collinear(CEA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(RadiusOfCircle(O),2)
EquilateralTriangle(ABC)
IsCentreOfCircle(O,O)
IsMidpointOfLine(D,AB)
IsMidpointOfLine(E,CA)
Goal: Value(LengthOfLine(DE))
Answer: sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
equilateral_triangle_property_angle(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
cosine_theorem(1,OBC)
midsegment_of_triangle_judgment_midpoint(1,DE,ABC)
midsegment_of_triangle_property_length(1,DE,ABC)
Problem: As shown in the diagram, DB=BO, ∠CAD=50°, the center of ⊙O is O. Find the measure of ∠DBO.
问题：如图所示，DB=BO，∠CAD=50°，O是圆O的圆心。求∠DBO的大小。
Construction (CDL):
Shape(AD,DB,OAB)
Shape(AD,DO,OC,CA)
Shape(DB,BO,OD)
Shape(CO,OB,OBC)
Shape(CA,OCA)
Collinear(ADB)
Collinear(DOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(DB),LengthOfLine(BO))
Equal(MeasureOfAngle(CAD),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBO))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
adjacent_complementary_angle(1,COB,BOD)
isosceles_triangle_judgment_line_equal(1,BOD)
isosceles_triangle_property_angle_equal(1,BOD)
triangle_property_angle_sum(1,DBO)
Problem: As shown in the diagram, ∠CAB=60°, the radius of ⊙O is 2, the center of circle O is O. 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。求直线BC的长度。
Construction (CDL):
Shape(AB,OAB)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(BC,OBC)
Shape(CA,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: 2*sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
cosine_theorem(1,OBC)
Problem: As shown in the diagram, ⌒OBA is congruent to ⌒ODC, ∠AOB=45°, the center of circle O is O. Find the measure of ∠COD.
问题：如图所示，⌒OBA与⌒ODC全等，∠AOB=45°，O是圆O的圆心。求∠COD的大小。
Construction (CDL):
Shape(AO,OD,OAD)
Shape(AO,OB,OBA)
Shape(BO,OC,OCB)
Shape(CO,OD,ODC)
Cocircular(O,ADCB)
Conditions (CDL):
CongruentBetweenArc(OBA,ODC)
Equal(MeasureOfAngle(AOB),45)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COD))
Answer: 45
Theorem sequence:
congruent_arc_property_measure_equal(1,OBA,ODC)
arc_property_center_angle(1,OBA,O)
arc_property_center_angle(1,ODC,O)
Problem: As shown in the diagram, ∠BCM=20°, the center of ⊙O is O, AO is parallel to BC. Find the measure of ∠BMA.
问题：如图所示，∠BCM=20°，O是圆O的圆心，AO∥BC。求∠BMA的大小。
Construction (CDL):
Shape(AO,OM,MC,OCA)
Shape(AM,MO,OA)
Shape(BM,MA,OAB)
Shape(MB,BC,CM)
Shape(CB,OBC)
Collinear(AMC)
Collinear(BMO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCM),20)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,BC)
Goal: Value(MeasureOfAngle(BMA))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
parallel_property_alternate_interior_angle(2,AO,BC)
triangle_property_angle_sum(1,MBC)
adjacent_complementary_angle(1,CMB,BMA)
Problem: As shown in the diagram, ⌒ODB is congruent to ⌒OBC, the diameter of circle O is equal to the length of line AB, ∠CAB=24°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠DBA.
问题：如图所示，⌒ODB全等于⌒OBC，圆O的直径与直线AB的长度相等，∠CAB=24°，O是⊙O的圆心，AB是⊙O的直径。求∠DBA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,OBC)
Shape(BO,OA,AD,DB)
Shape(BD,ODB)
Shape(OAD,DA)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
CongruentBetweenArc(ODB,OBC)
Equal(DiameterOfCircle(O),LengthOfLine(AB))
Equal(MeasureOfAngle(CAB),24)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 66
Theorem sequence:
congruent_arc_property_measure_equal(1,ODB,OBC)
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,OBC,A)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,BAD)
Problem: As shown in the diagram, ⌒OBC is congruent to ⌒OCD, ⌒OCD is congruent to ⌒ODE, the diameter of circle O is equal to the length of line AB, ∠DOC=35°, the center of circle O is O. Find the measure of ∠AOE.
问题：如图所示，弧OBC与弧OCD全等，⌒OCD与⌒ODE全等，⊙O的直径与直线AB的长度相等，∠DOC=35°，圆O的圆心为O。求∠AOE的大小。
Construction (CDL):
Shape(AO,OE,OEA)
Shape(EO,OD,ODE)
Shape(DO,OC,OCD)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCDE)
Conditions (CDL):
CongruentBetweenArc(OBC,OCD)
CongruentBetweenArc(OCD,ODE)
Equal(DiameterOfCircle(O),LengthOfLine(AB))
Equal(MeasureOfAngle(DOC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 75
Theorem sequence:
congruent_arc_property_measure_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OCD,ODE)
arc_property_center_angle(1,OBC,O)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,ODE,O)
angle_addition(1,EOD,DOC)
angle_addition(1,EOC,COB)
adjacent_complementary_angle(1,AOE,EOB)
Problem: As shown in the diagram, OA=EC, ∠BOA=120°, the radius of circle O is 25, the center of ⊙O is O, AC is perpendicular to BC, OA⊥CA, OE⊥BE, OACE is a rectangle. Find the length of line BC.
问题：如图所示，OA=EC，∠BOA=120°，⊙O的半径为25，O是圆O的圆心，AC垂直于BC，OA垂直于CA，OE⊥BE，OACE是长方形。求直线BC的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BO,OE,EB)
Shape(DE,EO,OA,OAD)
Shape(AC,CD,OAD)
Shape(BE,ED,ODB)
Collinear(BEDC)
Cocircular(O,ADB)
Conditions (CDL):
Equal(LengthOfLine(OA),LengthOfLine(EC))
Equal(MeasureOfAngle(BOA),120)
Equal(RadiusOfCircle(O),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(OA,CA)
PerpendicularBetweenLine(OE,BE)
Rectangle(OACE)
Goal: Value(LengthOfLine(BC))
Answer: 75/2
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
angle_addition(1,BOE,EOA)
sine_theorem(1,BOE)
line_addition(1,BE,EC)
Problem: As shown in the diagram, ∠OBC=40°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=40°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠AOB=90°, the radius of circle O is 2, O is the center of ⊙O. Find the area of △AOB.
问题：如图所示，∠AOB=90°，圆O的半径为2，O是⊙O的圆心。求三角形AOB的面积。
Construction (CDL):
Shape(AB,OBA)
Shape(AO,OB,BA)
Shape(BO,OA,OAB)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),90)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
Goal: Value(AreaOfTriangle(AOB))
Answer: 2
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
triangle_area_formula_sine(1,OBA)
Problem: As shown in the diagram, ∠OMN=50°, the center of circle O is O. Find the measure of ∠NOM.
问题：如图所示，∠OMN=50°，圆O的圆心为O。求∠NOM的大小。
Construction (CDL):
Shape(MO,ON,ONM)
Shape(OM,MN,NO)
Shape(NM,OMN)
Cocircular(O,MN)
Conditions (CDL):
Equal(MeasureOfAngle(OMN),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(NOM))
Answer: 80
Theorem sequence:
radius_of_circle_property_length_equal(1,OM,O)
radius_of_circle_property_length_equal(1,ON,O)
isosceles_triangle_judgment_line_equal(1,OMN)
isosceles_triangle_property_angle_equal(1,OMN)
triangle_property_angle_sum(1,OMN)
Problem: As shown in the diagram, ⌒OBC is congruent to ⌒OCD, ⌒OCD is congruent to ⌒ODE, the diameter of circle O is equal to the length of line AB, ∠DOC=40°, O is the center of circle O. Find the measure of ∠AOE.
问题：如图所示，⌒OBC全等于⌒OCD，⌒OCD与⌒ODE全等，⊙O的直径与直线AB的长度相等，∠DOC=40°，⊙O的圆心为O。求∠AOE的大小。
Construction (CDL):
Shape(AO,OE,OEA)
Shape(EO,OD,ODE)
Shape(DO,OC,OCD)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCDE)
Conditions (CDL):
CongruentBetweenArc(OBC,OCD)
CongruentBetweenArc(OCD,ODE)
Equal(DiameterOfCircle(O),LengthOfLine(AB))
Equal(MeasureOfAngle(DOC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 60
Theorem sequence:
congruent_arc_property_measure_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OCD,ODE)
arc_property_center_angle(1,OBC,O)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,ODE,O)
angle_addition(1,EOD,DOC)
angle_addition(1,EOC,COB)
adjacent_complementary_angle(1,AOE,EOB)
Problem: As shown in the diagram, ∠ABO=37°, the center of circle O is O. Find the measure of ⌒OAB.
问题：如图所示，∠ABO=37°，⊙O的圆心为O。求⌒OAB的角度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BO,OA,AB)
Shape(BA,OAB)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),37)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfArc(OAB))
Answer: 106
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,BOA)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, the measure of ⌒ODE is 50, O is the center of circle O, ED∥AB. Find the measure of ∠BOC.
问题：如图所示，弧ODE的角度为50，⊙O的圆心为O，ED平行于AB。求∠BOC的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(AO,OE,OEA)
Shape(EO,OD,DE)
Shape(ED,ODE)
Shape(DO,OB,OBD)
Shape(BO,OC,OCB)
Collinear(AOB)
Collinear(COD)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(MeasureOfArc(ODE),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(ED,AB)
Goal: Value(MeasureOfAngle(BOC))
Answer: 115
Theorem sequence:
arc_property_center_angle(1,ODE,O)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OE,O)
isosceles_triangle_judgment_line_equal(1,ODE)
isosceles_triangle_property_angle_equal(1,ODE)
triangle_property_angle_sum(1,EOD)
parallel_property_collinear_extend(3,BA,DE,O)
parallel_property_alternate_interior_angle(1,OA,DE)
angle_addition(1,AOE,EOD)
vertical_angle(1,AOD,BOC)
Problem: As shown in the diagram, ∠CAB=60°, the radius of ⊙O is 2, OD bisects ∠COB, the center of ⊙O is O, OD perpendicular bisects BC. Find the length of line OD.
问题：如图所示，∠CAB=60°，圆O的半径为2，OD是∠COB的角平分线，圆O的圆心为O，OD垂直平分AB。求直线OD的长度。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(EB,BD,DF,FE)
Shape(OE,EF,FO)
Shape(AE,EO,OC,OCA)
Shape(OF,FC,CO)
Shape(FD,DC,CF)
Shape(CD,DB,OBC)
Collinear(AEFC)
Collinear(OEB)
Collinear(OFD)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
Equal(RadiusOfCircle(O),2)
IsBisectorOfAngle(OD,COB)
IsCentreOfCircle(O,O)
IsPerpendicularBisectorOfLine(OD,BC)
Goal: Value(LengthOfLine(OD))
Answer: 1
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
cosine_theorem(1,BCO)
cosine_theorem(1,OBC)
sine_theorem(1,BDO)
sine_theorem(1,DOB)
Problem: As shown in the diagram, ∠OAB=50°, the center of circle O is O, C bisects segment AB. Find the measure of ∠BOC.
问题：如图所示，∠OAB=50°，O是⊙O的圆心，C是线段AB的中点。求∠BOC的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Shape(BA,OAB)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),50)
IsCentreOfCircle(O,O)
IsMidpointOfLine(C,AB)
Goal: Value(MeasureOfAngle(BOC))
Answer: 40
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,OCB)
median_of_triangle_judgment(1,OC,OAB)
isosceles_triangle_property_line_coincidence(2,OAB,C)
Problem: As shown in the diagram, ∠COA=70°, O is the center of circle O, AD∥CO. Find the measure of ∠AOD.
问题：如图所示，∠COA=70°，O是⊙O的圆心，AD平行于CO。求∠AOD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(DO,OB,OBD)
Shape(OA,OAC,CO)
Shape(BO,OC,OCB)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),70)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(AOD))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,CO)
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,DAO)
Problem: As shown in the diagram, ∠BAO=30°, O is the center of ⊙O. Find the measure of ∠AOB.
问题：如图所示，∠BAO=30°，O是⊙O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OB)
Shape(BO,OA,OAB)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 120
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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,BAO)
Problem: As shown in the diagram, ∠BAC=30°, the center of ⊙O is O, △ACB is a right △. Find the measure of ⌒ODA.
问题：如图所示，∠BAC=30°，⊙O的圆心为O，AC和CB是直角三角形ACB的两个直角边。求⌒ODA的角度。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(DA,AO,OD)
Shape(AD,ODA)
Shape(DO,OC,OCD)
Shape(CB,BD,OCD)
Collinear(AOC)
Collinear(ADB)
Cocircular(O,ACD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
RightTriangle(ACB)
Goal: Value(MeasureOfArc(ODA))
Answer: 120
Theorem sequence:
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,DAO)
arc_property_center_angle(1,ODA,O)