SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=AE, ∠EAB=30°, ∠GAD=30°, quadrilateral ABCD is a parallelogram, AG and EF are opposite sides of the parallelogram AEFG. Find the measure of ∠BCD.
问题：如图所示，AB=AE，∠EAB=30°，∠GAD=30°，BA和CD是▱ABCD的一组对边，AEFG是▱。求∠BCD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EC,CH,HE)
Shape(AE,EH,HD,DA)
Shape(AD,DG,GA)
Shape(FD,DH,HF)
Collinear(BEC)
Collinear(EHF)
Collinear(FDG)
Collinear(CHD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AE))
Equal(MeasureOfAngle(EAB),30)
Equal(MeasureOfAngle(GAD),30)
Parallelogram(ABCD)
Parallelogram(AEFG)
Goal: Value(MeasureOfAngle(BCD))
Answer: 105
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABE)
isosceles_triangle_property_angle_equal(1,ABE)
triangle_property_angle_sum(1,ABE)
parallel_property_ipsilateral_internal_angle(1,BA,CD)
Problem: As shown in the diagram, BD=3, the perimeter of triangle AEC is 20, ED is the perpendicular bisector of BC. Find the perimeter of △ABC.
问题：如图所示，BD=3，三角形AEC的周长为20，ED垂直平分AB。求△ABC的周长。
Construction (CDL):
Shape(AE,EC,CA)
Shape(CE,ED,DC)
Shape(DE,EB,BD)
Collinear(AEB)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(BD),3)
Equal(PerimeterOfTriangle(AEC),20)
IsPerpendicularBisectorOfLine(ED,BC)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 26
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,ED,BC)
line_addition(1,AE,EB)
line_addition(1,CD,DB)
triangle_perimeter_formula(1,AEC)
triangle_perimeter_formula(1,CAB)
Problem: As shown in the diagram, ∠BCA=50°, ∠BDC=90°, ∠DCB=25°. Find the measure of ∠ECD.
问题：如图所示，∠BCA=50°，∠BDC=90°，∠DCB=25°。求∠ECD的大小。
Construction (CDL):
Shape(BC,CA,AB)
Shape(BD,DC,CB)
Shape(EC,CD)
Collinear(ECA)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
Equal(MeasureOfAngle(BDC),90)
Equal(MeasureOfAngle(DCB),25)
Goal: Value(MeasureOfAngle(ECD))
Answer: 105
Theorem sequence:
angle_addition(1,DCB,BCA)
adjacent_complementary_angle(1,ECD,DCA)
Problem: As shown in the diagram, MN=40, M is the midpoint of segment AC, N is the midpoint of segment CB. Find the length of line AB.
问题：如图所示，MN=40，M是线段AC的中点，N平分线段CB。求直线AB的长度。
Construction (CDL):
Shape(CN,NM,MC)
Shape(MN,NB,BA,AM)
Collinear(CMA)
Collinear(CNB)
Conditions (CDL):
Equal(LengthOfLine(MN),40)
IsMidpointOfLine(M,AC)
IsMidpointOfLine(N,CB)
Goal: Value(LengthOfLine(AB))
Answer: 80
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,NM,CBA)
midsegment_of_triangle_property_length(1,NM,CBA)
Problem: As shown in the diagram, ∠BDF=60°, ∠GFH=25°, BC∥DE, DE∥FG. Find the measure of ∠ABC.
问题：如图所示，∠BDF=60°，∠GFH=25°，BC平行于DE，DE∥FG。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC)
Shape(CB,BD)
Shape(BD,DE)
Shape(ED,DF)
Shape(DF,FG)
Shape(GF,FH)
Collinear(ABD)
Collinear(DFH)
Conditions (CDL):
Equal(MeasureOfAngle(BDF),60)
Equal(MeasureOfAngle(GFH),25)
ParallelBetweenLine(BC,DE)
ParallelBetweenLine(DE,FG)
Goal: Value(MeasureOfAngle(ABC))
Answer: 35
Theorem sequence:
angle_addition(1,BDE,EDF)
parallel_property_corresponding_angle(1,BC,DE,A)
parallel_property_corresponding_angle(2,DE,FG,H)
Problem: As shown in the diagram, AB=AC, BC=12, DC=4, DC=EF, FC=7, DC is parallel to EF. Find the perimeter of △EBF.
问题：如图所示，AB=AC，BC=12，DC=4，DC=EF，FC=7，DC∥EF。求三角形EBF的周长。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BF,FE)
Shape(DE,EF,FC,CD)
Collinear(BFC)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(BC),12)
Equal(LengthOfLine(DC),4)
Equal(LengthOfLine(DC),LengthOfLine(EF))
Equal(LengthOfLine(FC),7)
ParallelBetweenLine(DC,EF)
Goal: Value(PerimeterOfTriangle(EBF))
Answer: 13
Theorem sequence:
line_addition(1,BF,FC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
parallel_property_corresponding_angle(1,FE,CD,B)
isosceles_triangle_judgment_angle_equal(1,EBF)
triangle_perimeter_formula(1,EBF)
Problem: As shown in the diagram, ∠CAB=50°, ∠ECA=90°, △ECD is mirror congruent to △ADC. Find the measure of ∠EDB.
问题：如图所示，∠CAB=50°，∠ECA=90°，三角形ECD镜像全等于三角形ADC。求∠EDB的大小。
Construction (CDL):
Shape(BE,ED,DB)
Shape(EC,CD,DE)
Shape(DC,CA,AD)
Collinear(BDA)
Collinear(BEC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
Equal(MeasureOfAngle(ECA),90)
MirrorCongruentBetweenTriangle(ECD,ADC)
Goal: Value(MeasureOfAngle(EDB))
Answer: 10
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,ECD,ADC)
angle_addition(1,ECD,DCA)
angle_addition(1,ADC,CDE)
adjacent_complementary_angle(1,ADE,EDB)
triangle_property_angle_sum(1,ECD)
triangle_property_angle_sum(1,DCA)
Problem: As shown in the diagram, ∠COE=140°, OE is the angle bisector of ∠AOD. Find the measure of ∠BOC.
问题：如图所示，∠COE=140°，OE是∠AOD的角平分线。求∠BOC的大小。
Construction (CDL):
Shape(BO,OC)
Shape(CO,OA)
Shape(AO,OE)
Shape(EO,OD)
Shape(DO,OB)
Collinear(BOA)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(COE),140)
IsBisectorOfAngle(OE,AOD)
Goal: Value(MeasureOfAngle(BOC))
Answer: 80
Theorem sequence:
angle_addition(1,COA,AOE)
adjacent_complementary_angle(1,BOC,COA)
adjacent_complementary_angle(1,COE,EOD)
Problem: As shown in the diagram, AC=21, BC=32, DE=18, D bisects segment AC, E is the midpoint of segment CB. Find the length of line AB.
问题：如图所示，AC=21，BC=32，DE=18，D是线段AC的中点，E是线段CB的中点。求直线AB的长度。
Construction (CDL):
Shape(CE,ED,DC)
Shape(AD,DE,EB,BA)
Collinear(CDA)
Collinear(CEB)
Conditions (CDL):
Equal(LengthOfLine(AC),21)
Equal(LengthOfLine(BC),32)
Equal(LengthOfLine(DE),18)
IsMidpointOfLine(D,AC)
IsMidpointOfLine(E,CB)
Goal: Value(LengthOfLine(AB))
Answer: 36
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,ED,CBA)
midsegment_of_triangle_property_length(1,ED,CBA)
Problem: As shown in the diagram, AC=6, the perimeter of △ABD is 13, DE perpendicular bisects AC, DE is perpendicular to AE. Find the perimeter of triangle ABC.
问题：如图所示，AC=6，△ABD的周长为13，DE垂直平分AB，DE⊥AE。求△ABC的周长。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(PerimeterOfTriangle(ABD),13)
IsPerpendicularBisectorOfLine(DE,AC)
PerpendicularBetweenLine(DE,AE)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 19
Theorem sequence:
line_addition(1,BD,DC)
perpendicular_bisector_property_distance_equal(1,DE,AC)
triangle_perimeter_formula(1,ABD)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, ∠AED=26°, AE is the angle bisector of ∠DAB, DE is parallel to AB, quadrilateral ABCD is a parallelogram. Find the measure of ∠BCE.
问题：如图所示，∠AED=26°，AE是∠DAB的角平分线，DE∥AB，ABCD是平行四边形。求∠BCE的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AB,BC,CE)
Collinear(DEC)
Conditions (CDL):
Equal(MeasureOfAngle(AED),26)
IsBisectorOfAngle(AE,DAB)
ParallelBetweenLine(DE,AB)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BCE))
Answer: 52
Theorem sequence:
parallel_property_alternate_interior_angle(2,DE,AB)
angle_addition(1,DEA,EAB)
parallelogram_property_opposite_angle_equal(1,ABCD)
Problem: As shown in the diagram, ∠EBA=120°, ∠ECD=25°, AB is parallel to CD. Find the measure of ∠CEB.
问题：如图所示，∠EBA=120°，∠ECD=25°，AB平行于CD。求∠CEB的大小。
Construction (CDL):
Shape(EC,CD,DE)
Shape(AC,CE,EB,BA)
Collinear(BED)
Conditions (CDL):
Equal(MeasureOfAngle(EBA),120)
Equal(MeasureOfAngle(ECD),25)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(CEB))
Answer: 85
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,DC,BA)
triangle_property_angle_sum(1,ECD)
adjacent_complementary_angle(1,DEC,CEB)
Problem: As shown in the diagram, ∠COF=70°, CO⊥BO. Find the measure of ∠EOA.
问题：如图所示，∠COF=70°，CO⊥BO。求∠EOA的大小。
Construction (CDL):
Shape(CO,OF)
Shape(FO,OB)
Shape(BO,OD)
Shape(DO,OE)
Shape(EO,OA)
Shape(AO,OC)
Collinear(AOB)
Collinear(COD)
Collinear(EOF)
Conditions (CDL):
Equal(MeasureOfAngle(COF),70)
PerpendicularBetweenLine(CO,BO)
Goal: Value(MeasureOfAngle(EOA))
Answer: 20
Theorem sequence:
angle_addition(1,COF,FOB)
vertical_angle(1,EOA,FOB)
Problem: As shown in the diagram, ∠BOC=30°, BO⊥AO. Find the measure of ∠COA.
问题：如图所示，∠BOC=30°，BO垂直于AO。求∠COA的大小。
Construction (CDL):
Shape(CO,OA)
Shape(BO,OC)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),30)
PerpendicularBetweenLine(BO,AO)
Goal: Value(MeasureOfAngle(COA))
Answer: 60
Theorem sequence:
angle_addition(1,BOC,COA)
Problem: As shown in the diagram, AB=AC, ∠EAD=40°, ED perpendicular bisects AB. Find the measure of ∠EBC.
问题：如图所示，AB=AC，∠EAD=40°，ED是AB的垂直平分线。求∠EBC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BE,ED)
Shape(EB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(EAD),40)
IsPerpendicularBisectorOfLine(ED,AB)
Goal: Value(MeasureOfAngle(EBC))
Answer: 30
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,ED,AB)
isosceles_triangle_judgment_line_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
isosceles_triangle_judgment_line_equal(1,EAB)
isosceles_triangle_property_angle_equal(1,EAB)
angle_addition(1,DBE,EBC)
Problem: As shown in the diagram, ∠DBA=22°, the center of circle O is O, AB is parallel to CO. Find the measure of ∠BAD.
问题：如图所示，∠DBA=22°，O是⊙O的圆心，AB平行于CO。求∠BAD的大小。
Construction (CDL):
Shape(BA,AD,DB)
Shape(DC,CO,OD)
Shape(CD,DA,OAC)
Shape(AB,OBA)
Shape(BD,DO,OC,OCB)
Collinear(BDC)
Collinear(ADO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(DBA),22)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CO)
Goal: Value(MeasureOfAngle(BAD))
Answer: 44
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
parallel_property_alternate_interior_angle(1,AB,CO)
Problem: As shown in the diagram, ∠ACD=55°, BA is parallel to CD. Find the measure of ∠CAB.
问题：如图所示，∠ACD=55°，BA∥CD。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD)
Shape(DC,CE)
Collinear(BCE)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),55)
ParallelBetweenLine(BA,CD)
Goal: Value(MeasureOfAngle(CAB))
Answer: 55
Theorem sequence:
parallel_property_alternate_interior_angle(2,BA,CD)
Problem: As shown in the diagram, DA=DF, ∠FDA=40°, DC and AB are opposite sides of the ▱ DABC, DFGH is a parallelogram. Find the measure of ∠FBC.
问题：如图所示，DA=DF，∠FDA=40°，四边形DABC是平行四边形，DH和FG是▱DFGH的一组对边。求∠FBC的大小。
Construction (CDL):
Shape(EF,FB,BC,CE)
Shape(DF,FE,ED)
Shape(DA,AF,FD)
Shape(DE,EG,GH,HD)
Shape(DE,EC)
Collinear(DEC)
Collinear(AFB)
Collinear(FEG)
Conditions (CDL):
Equal(LengthOfLine(DA),LengthOfLine(DF))
Equal(MeasureOfAngle(FDA),40)
Parallelogram(DABC)
Parallelogram(DFGH)
Goal: Value(MeasureOfAngle(FBC))
Answer: 110
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DAF)
isosceles_triangle_property_angle_equal(1,DAF)
triangle_property_angle_sum(1,DAF)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
Problem: As shown in the diagram, ∠BCN=∠NCQ, ∠MBA=80°, CP∥BM. Find the measure of ∠NCQ.
问题：如图所示，∠BCN=∠NCQ，∠MBA=80°，CP平行于BM。求∠NCQ的大小。
Construction (CDL):
Shape(MB,BA)
Shape(AB,BN)
Shape(CB,BM)
Shape(NB,BC)
Shape(RN,NC)
Shape(CN,NB)
Shape(PC,CB)
Shape(BC,CN)
Shape(NC,CQ)
Shape(BC,CN,NB)
Collinear(ABC)
Collinear(MBNR)
Collinear(PCQ)
Conditions (CDL):
Equal(MeasureOfAngle(BCN),MeasureOfAngle(NCQ))
Equal(MeasureOfAngle(MBA),80)
ParallelBetweenLine(CP,BM)
Goal: Value(MeasureOfAngle(NCQ))
Answer: 50
Theorem sequence:
parallel_property_corresponding_angle(2,CP,BM,A)
angle_addition(1,PCB,BCN)
angle_addition(1,PCN,NCQ)
flat_angle(1,PCQ)
Problem: As shown in the diagram, ∠BOC=50°, OD is the angle bisector of ∠AOE. Find the measure of ∠EOB.
问题：如图所示，∠BOC=50°，OD是∠AOE的角平分线。求∠EOB的大小。
Construction (CDL):
Shape(AO,OD)
Shape(DO,OE)
Shape(EO,OB)
Shape(BO,OC)
Shape(CO,OA)
Collinear(AOB)
Collinear(DOC)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsBisectorOfAngle(OD,AOE)
Goal: Value(MeasureOfAngle(EOB))
Answer: 80
Theorem sequence:
vertical_angle(1,BOC,AOD)
angle_addition(1,AOE,DOE)
angle_addition(1,AOE,EOB)
flat_angle(1,AOB)
Problem: As shown in the diagram, AC=12, BC=9, BC is perpendicular to AC. Find the sum of the length of line CB and the length of line BA.
问题：如图所示，AC=12，BC=9，BC⊥AC。求直线CB的长度与直线BA的长度之和。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),9)
PerpendicularBetweenLine(BC,AC)
Goal: Value(Add(LengthOfLine(CB),LengthOfLine(BA)))
Answer: 24
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, AD=2, CF=2, the perimeter of △ABC is 16, AD is parallel to CF. Find the perimeter of ABFD.
问题：如图所示，AD=2，CF=2，三角形ABC的周长为16，AD∥CF。求ABFD的周长。
Construction (CDL):
Shape(AG,GD,DA)
Shape(AB,BE,EG,GA)
Shape(GE,EC,CG)
Shape(DG,GC,CF,FD)
Collinear(AGC)
Collinear(DGE)
Collinear(BECF)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(CF),2)
Equal(PerimeterOfTriangle(ABC),16)
ParallelBetweenLine(AD,CF)
Goal: Value(PerimeterOfQuadrilateral(ABFD))
Answer: 20
Theorem sequence:
line_addition(1,BC,CF)
triangle_perimeter_formula(1,ABC)
parallelogram_judgment_parallel_and_equal(1,DACF)
parallelogram_property_opposite_line_equal(1,ACFD)
quadrilateral_perimeter_formula(1,ABFD)
Problem: As shown in the diagram, ∠CAD=50°, AO is the angle bisector of ∠CAD, the center of circle O is O, AD is perpendicular to OD, OD is perpendicular to BD. Find the measure of ∠BCO.
问题：如图所示，∠CAD=50°，AO平分∠CAD，圆O的圆心为O，AD垂直于OD，OD垂直于BD。求∠BCO的大小。
Construction (CDL):
Shape(CA,AO,OC)
Shape(AD,DO,OA)
Shape(CO,OB,BC)
Shape(OD,DB,BO)
Shape(AC,OCA)
Shape(BD,DA,OAB)
Shape(CB,OBC)
Collinear(ADB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),50)
IsBisectorOfAngle(AO,CAD)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
PerpendicularBetweenLine(OD,BD)
Goal: Value(MeasureOfAngle(BCO))
Answer: 40
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)
angle_addition(1,CAO,OAD)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OCA)
mirror_congruent_triangle_judgment_aas(1,OAD,ODB)
mirror_congruent_triangle_property_angle_equal(1,OAD,ODB)
triangle_property_angle_sum(1,OAD)
angle_addition(1,BOD,DOA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
angle_addition(1,BCO,OCA)
Problem: As shown in the diagram, ∠AFG=22°, ∠EFG=45°, AF is parallel to EB. Find the measure of ∠FEB.
问题：如图所示，∠AFG=22°，∠EFG=45°，AF平行于EB。求∠FEB的大小。
Construction (CDL):
Shape(GA,AF,FG)
Shape(AE,EF,FA)
Shape(FE,EB)
Shape(DE,EA)
Shape(EA,AX)
Shape(XA,AG)
Shape(GF,FC)
Shape(CF,FE)
Collinear(GAE)
Collinear(XAFG)
Collinear(DEB)
Conditions (CDL):
Equal(MeasureOfAngle(AFG),22)
Equal(MeasureOfAngle(EFG),45)
ParallelBetweenLine(AF,EB)
Goal: Value(MeasureOfAngle(FEB))
Answer: 23
Theorem sequence:
angle_addition(1,EFA,AFG)
parallel_property_alternate_interior_angle(2,AF,EB)
Problem: As shown in the diagram, ∠BOF=42°, EC is parallel to OA, HO is perpendicular to GO. Find the measure of ∠GEC.
问题：如图所示，∠BOF=42°，EC平行于OA，HO垂直于GO。求∠GEC的大小。
Construction (CDL):
Shape(OE,EF,FO)
Shape(EO,OA)
Shape(CE,EO)
Shape(GE,EC)
Shape(FE,EG)
Shape(HF,FE)
Shape(DF,FH)
Shape(OF,FD)
Shape(BO,OF)
Collinear(AOB)
Collinear(CEFD)
Collinear(OEG)
Collinear(OFH)
Conditions (CDL):
Equal(MeasureOfAngle(BOF),42)
ParallelBetweenLine(EC,OA)
PerpendicularBetweenLine(HO,GO)
Goal: Value(MeasureOfAngle(GEC))
Answer: 48
Theorem sequence:
angle_addition(1,BOF,FOE)
angle_addition(1,BOE,EOA)
flat_angle(1,BOA)
parallel_property_ipsilateral_internal_angle(1,EC,OA)
adjacent_complementary_angle(1,GEC,CEO)
Problem: As shown in the diagram, OB=OA, ∠OBA=25°, the center of circle O is O, the tangent to circle O is DA. Find the measure of ∠ADC.
问题：如图所示，OB=OA，∠OBA=25°，O是圆O的圆心，圆O的切线为DA。求∠ADC的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(OB,BA,AO)
Shape(BO,OC,OCB)
Shape(AB,OBA)
Shape(AD,DC,OAC)
Collinear(BOCD)
Cocircular(O,BAC)
Conditions (CDL):
Equal(LengthOfLine(OB),LengthOfLine(OA))
Equal(MeasureOfAngle(OBA),25)
IsCentreOfCircle(O,O)
IsTangentOfCircle(DA,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
tangent_of_circle_property_perpendicular(1,DA,O,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
angle_addition(1,BAO,OAD)
triangle_property_angle_sum(1,DBA)
Problem: As shown in the diagram, ∠AMN=36°, ∠ANT=140°, ∠AQP=36°. Find the measure of ∠QPA.
问题：如图所示，∠AMN=36°，∠ANT=140°，∠AQP=36°。求∠QPA的大小。
Construction (CDL):
Shape(PA,AQ,QP)
Shape(AM,MN,NA)
Shape(AN,NT)
Collinear(MAQ)
Collinear(PAN)
Collinear(MNT)
Conditions (CDL):
Equal(MeasureOfAngle(AMN),36)
Equal(MeasureOfAngle(ANT),140)
Equal(MeasureOfAngle(AQP),36)
Goal: Value(MeasureOfAngle(QPA))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,MNA,ANT)
triangle_property_angle_sum(1,NAM)
vertical_angle(1,PAQ,NAM)
triangle_property_angle_sum(1,PAQ)
Problem: As shown in the diagram, ∠ACB=40°, ∠BAC=90°, BE is the angle bisector of ∠ABD. Find the measure of ∠EBD.
问题：如图所示，∠ACB=40°，∠BAC=90°，BE是∠ABD的角平分线。求∠EBD的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(AB,BE)
Shape(EB,BD)
Collinear(CBD)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),40)
Equal(MeasureOfAngle(BAC),90)
IsBisectorOfAngle(BE,ABD)
Goal: Value(MeasureOfAngle(EBD))
Answer: 65
Theorem sequence:
triangle_property_angle_sum(1,ACB)
angle_addition(1,CBA,ABE)
angle_addition(1,CBE,EBD)
flat_angle(1,CBD)
Problem: As shown in the diagram, ∠DCE=30°, ∠PBA=38°, ED is parallel to BQ, AB⊥DB. Find the measure of ∠AEM.
问题：如图所示，∠DCE=30°，∠PBA=38°，ED∥BQ，AB垂直于DB。求∠AEM的大小。
Construction (CDL):
Shape(CE,ED,DC)
Shape(EA,AB,BD,DE)
Shape(ME,EC)
Shape(CD,DN)
Shape(AE,EM)
Shape(ND,DB)
Shape(DB,BQ)
Shape(PB,BA)
Collinear(MEDN)
Collinear(PBQ)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),30)
Equal(MeasureOfAngle(PBA),38)
ParallelBetweenLine(ED,BQ)
PerpendicularBetweenLine(AB,DB)
Goal: Value(MeasureOfAngle(AEM))
Answer: 22
Theorem sequence:
angle_addition(1,PBA,ABD)
angle_addition(1,PBD,DBQ)
flat_angle(1,PBQ)
parallel_property_alternate_interior_angle(2,ED,BQ)
adjacent_complementary_angle(1,BDE,EDC)
triangle_property_angle_sum(1,EDC)
vertical_angle(1,AEM,CED)
Problem: As shown in the diagram, OD=1/2, O is the midpoint of segment AB, AC⊥DC, OD⊥BD. Find the length of line AC.
问题：如图所示，OD=1/2，O是线段AB的中点，AC⊥DC，OD垂直于BD。求直线AC的长度。
Construction (CDL):
Shape(OD,DB,BO)
Shape(AC,CD,DO,OA)
Collinear(AOB)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(OD),1/2)
IsMidpointOfLine(O,AB)
PerpendicularBetweenLine(AC,DC)
PerpendicularBetweenLine(OD,BD)
Goal: Value(LengthOfLine(AC))
Answer: 1
Theorem sequence:
line_addition(1,AO,OB)
similar_triangle_judgment_aa(1,ODB,ACB)
similar_triangle_property_line_ratio(1,DBO,CBA)
similar_triangle_property_line_ratio(1,BOD,BAC)
Problem: As shown in the diagram, ∠CBA=60°, ∠EDC=150°, FG∥DE. Find the measure of ∠BCF.
问题：如图所示，∠CBA=60°，∠EDC=150°，FG∥DE。求∠BCF的大小。
Construction (CDL):
Shape(BC,CD,DF,FB)
Shape(CB,BA)
Shape(ED,DC)
Shape(GF,FD)
Shape(FD,DE)
Collinear(ABFG)
Collinear(CDF)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),60)
Equal(MeasureOfAngle(EDC),150)
ParallelBetweenLine(FG,DE)
Goal: Value(MeasureOfAngle(BCF))
Answer: 30
Theorem sequence:
parallel_property_corresponding_angle(2,FG,DE,C)
adjacent_complementary_angle(1,GFD,DFB)
adjacent_complementary_angle(1,FBC,CBA)
triangle_property_angle_sum(1,BCF)
Problem: As shown in the diagram, ∠DBO=20°, the center of circle O is O. Find the measure of ∠BCE.
问题：如图所示，∠DBO=20°，O是⊙O的圆心。求∠BCE的大小。
Construction (CDL):
Shape(AE,EC,CA)
Shape(CE,EO,OB,BC)
Shape(AD,DE,EA)
Shape(ED,DO,OE)
Shape(OD,DB,BO)
Shape(DA,OAD)
Shape(BD,ODB)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBO),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCE))
Answer: 70
Theorem sequence:
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)
triangle_property_angle_sum(1,ODB)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠APB=70°, O is the center of circle O, the tangent to circle O is PA, the tangent to ⊙O is PB. Find the measure of ∠BCA.
问题：如图所示，∠APB=70°，⊙O的圆心为O，PA是⊙O的切线，圆O的切线为PB。求∠BCA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BO,OA,OAB)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(AO,OB,BC,CA)
Cocircular(O,BCA)
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, ∠ACB=56°, the center of ⊙O is O, the tangent to circle O is CA, CB is the tangent to circle O. Find the measure of ∠BDA.
问题：如图所示，∠ACB=56°，O是⊙O的圆心，CA是⊙O的切线，圆O的切线为CB。求∠BDA的大小。
Construction (CDL):
Shape(AC,CB,OAB)
Shape(BO,OA,OAB)
Shape(AD,ODA)
Shape(DB,OBD)
Shape(AO,OB,BD,DA)
Cocircular(O,BDA)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),56)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CA,O)
IsTangentOfCircle(CB,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 62
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CA,O,O)
tangent_of_circle_property_perpendicular(2,CB,O,O)
quadrilateral_property_angle_sum(1,ACBO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠ECB=40°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠ABD.
问题：如图所示，∠ECB=40°，圆O的圆心为O，AB是⊙O的直径。求∠ABD的大小。
Construction (CDL):
Shape(AC,CO,OA)
Shape(OC,CE,EO)
Shape(EC,CB,BE)
Shape(AO,OE,ED,DA)
Shape(DE,EB,BD)
Shape(AD,ODA)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(DB,OBD)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ECB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,A)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠DAB=80°, ∠EBC=25°, AD is parallel to BF. Find the measure of ∠CBA.
问题：如图所示，∠DAB=80°，∠EBC=25°，AD∥BF。求∠CBA的大小。
Construction (CDL):
Shape(DA,AB)
Shape(AB,BF)
Shape(CB,BA)
Shape(EB,BC)
Collinear(FBE)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),80)
Equal(MeasureOfAngle(EBC),25)
ParallelBetweenLine(AD,BF)
Goal: Value(MeasureOfAngle(CBA))
Answer: 55
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BF)
angle_addition(1,EBC,CBA)
angle_addition(1,EBA,ABF)
flat_angle(1,EBF)
Problem: As shown in the diagram, ∠EOC=30°, AO is perpendicular to EO. Find the measure of ∠BOD.
问题：如图所示，∠EOC=30°，AO⊥EO。求∠BOD的大小。
Construction (CDL):
Shape(AO,OE)
Shape(EO,OC)
Shape(CO,OB)
Shape(BO,OD)
Shape(DO,OA)
Collinear(AOB)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(EOC),30)
PerpendicularBetweenLine(AO,EO)
Goal: Value(MeasureOfAngle(BOD))
Answer: 120
Theorem sequence:
angle_addition(1,AOE,EOC)
vertical_angle(1,AOC,BOD)
Problem: As shown in the diagram, AC=6, BD=8, ABCD is a rhombus. Find the length of line AD.
问题：如图所示，AC=6，BD=8，四边形ABCD是菱形。求直线AD的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(BC,CO,OB)
Shape(OC,CD,DO)
Shape(AO,OD,DA)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BD),8)
Rhombus(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 5
Theorem sequence:
line_addition(1,AO,OC)
line_addition(1,BO,OD)
kite_property_diagonal_perpendicular_bisection(1,ABCD,O)
kite_property_diagonal_perpendicular_bisection(1,BCDA,O)
right_triangle_judgment_angle(1,AOD)
right_triangle_property_pythagorean(1,AOD)
Problem: As shown in the diagram, ∠DAP=60°, P is the midpoint of segment AB, AP is perpendicular to DP. Find the measure of ∠PDA.
问题：如图所示，∠DAP=60°，P平分线段AB，AP垂直于DP。求∠PDA的大小。
Construction (CDL):
Shape(DA,AP,PD)
Shape(EF,FB,BE)
Shape(PC,CF,FP)
Shape(DP,PF,FE,ED)
Collinear(APB)
Collinear(DPC)
Collinear(CFE)
Conditions (CDL):
Equal(MeasureOfAngle(DAP),60)
IsMidpointOfLine(P,AB)
PerpendicularBetweenLine(AP,DP)
Goal: Value(MeasureOfAngle(PDA))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,DAP)