SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CAB=70°, ∠DFE=70°. Find the sum of the measure of ∠CEF and the measure of ∠FDB.
问题：如图所示，∠CAB=70°，∠DFE=70°。求∠CEF的大小与∠FDB的大小之和。
Construction (CDL):
Shape(CE,EF,FD,DB,BC)
Shape(ED,DF,FE)
Shape(EA,AD,DE)
Collinear(CEA)
Collinear(BDA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),70)
Equal(MeasureOfAngle(DFE),70)
Goal: Value(Add(MeasureOfAngle(CEF),MeasureOfAngle(FDB)))
Answer: 140
Theorem sequence:
adjacent_complementary_angle(1,CEF,FEA)
adjacent_complementary_angle(1,ADF,FDB)
quadrilateral_property_angle_sum(1,EADF)
Problem: As shown in the diagram, AB is parallel to DC, BE is perpendicular to CE. Find the sum of the measure of ∠EBA and the measure of ∠FCE.
问题：如图所示，AB∥DC，BE⊥CE。求∠EBA的大小与∠FCE的大小之和。
Construction (CDL):
Shape(AD,DF,FE,EB,BA)
Shape(BE,EC,CB)
Shape(EF,FC,CE)
Collinear(DFC)
Collinear(BEF)
Conditions (CDL):
ParallelBetweenLine(AB,DC)
PerpendicularBetweenLine(BE,CE)
Goal: Value(Add(MeasureOfAngle(EBA),MeasureOfAngle(FCE)))
Answer: 90
Theorem sequence:
parallel_property_collinear_extend(3,CD,BA,F)
parallel_property_alternate_interior_angle(2,AB,FC)
adjacent_complementary_angle(1,BEC,CEF)
triangle_property_angle_sum(1,EFC)
Problem: As shown in the diagram, ∠DBE=40°, BE is parallel to CA, AB⊥CB. Find the measure of ∠CAB.
问题：如图所示，∠DBE=40°，BE平行于CA，AB垂直于CB。求∠CAB的大小。
Construction (CDL):
Shape(DB,BE)
Shape(EB,BA)
Shape(BC,CA,AB)
Collinear(DBC)
Conditions (CDL):
Equal(MeasureOfAngle(DBE),40)
ParallelBetweenLine(BE,CA)
PerpendicularBetweenLine(AB,CB)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
parallel_property_corresponding_angle(1,BE,CA,D)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, ∠CBD=20°, triangle ABC is an equilateral triangle, EG is parallel to BD. Find the measure of ∠GFC.
问题：如图所示，∠CBD=20°，三角形ABC为等边三角形，EG∥BD。求∠GFC的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BC,CF,FE)
Shape(CB,BD,DC)
Shape(AF,FG)
Shape(GF,FC)
Collinear(EFG)
Collinear(AEB)
Collinear(AFCD)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),20)
EquilateralTriangle(ABC)
ParallelBetweenLine(EG,BD)
Goal: Value(MeasureOfAngle(GFC))
Answer: 40
Theorem sequence:
equilateral_triangle_property_angle(1,CAB)
adjacent_complementary_angle(1,DCB,BCF)
triangle_property_angle_sum(1,CBD)
parallel_property_collinear_extend(3,EG,BD,F)
parallel_property_alternate_interior_angle(1,FG,BD)
Problem: As shown in the diagram, ∠COB=145°, DO is perpendicular to CO. Find the measure of ∠DOA.
问题：如图所示，∠COB=145°，DO⊥CO。求∠DOA的大小。
Construction (CDL):
Shape(AO,OC)
Shape(CO,OB)
Shape(BO,OD)
Shape(DO,OA)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(COB),145)
PerpendicularBetweenLine(DO,CO)
Goal: Value(MeasureOfAngle(DOA))
Answer: 55
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
angle_addition(1,DOA,AOC)
Problem: As shown in the diagram, △ABC is congruent to △EDC, AB=1, AC=2, AD=3, AC⊥EC, BC⊥DC. Find the measure of ∠CED.
问题：如图所示，三角形ABC与三角形EDC是全等三角形，AB=1，AC=2，AD=3，AC⊥EC，BC垂直于DC。求∠CED的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AF,FD)
Shape(FA,AC,CF)
Shape(DF,FE,ED)
Shape(FC,CE,EF)
Collinear(AFE)
Collinear(DFC)
Conditions (CDL):
CongruentBetweenTriangle(ABC,EDC)
Equal(LengthOfLine(AB),1)
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(AD),3)
PerpendicularBetweenLine(AC,EC)
PerpendicularBetweenLine(BC,DC)
Goal: Value(MeasureOfAngle(CED))
Answer: 135
Theorem sequence:
congruent_triangle_property_line_equal(1,BCA,DCE)
congruent_triangle_property_line_equal(1,CAB,CED)
right_triangle_judgment_angle(1,ACE)
right_triangle_property_pythagorean(1,ACE)
right_triangle_judgment_pythagorean_inverse(1,AED)
isosceles_triangle_judgment_line_equal(1,CEA)
isosceles_triangle_property_angle_equal(1,CEA)
triangle_property_angle_sum(1,CEA)
angle_addition(1,CEA,AED)
Problem: As shown in the diagram, ∠GED=80°, EF bisects ∠CEG, AB is parallel to CD. Find the measure of ∠GFE.
问题：如图所示，∠GED=80°，EF是∠CEG的角平分线，AB平行于CD。求∠GFE的大小。
Construction (CDL):
Shape(EF,FA)
Shape(BG,GE)
Shape(CE,EF)
Shape(GE,ED)
Shape(FE,EG,GF)
Collinear(AFGB)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(GED),80)
IsBisectorOfAngle(EF,CEG)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(GFE))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,CEG,GED)
angle_addition(1,CEF,FEG)
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,BF,E)
parallel_property_alternate_interior_angle(1,FB,CE)
Problem: As shown in the diagram, AC=4, DB=5, DABC is a rhombus. Find the area of DABC.
问题：如图所示，AC=4，DB=5，四边形DABC是菱形。求DABC的面积。
Construction (CDL):
Shape(DA,AE,ED)
Shape(DE,EC,CD)
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Collinear(AEC)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(DB),5)
Rhombus(DABC)
Goal: Value(AreaOfQuadrilateral(DABC))
Answer: 10
Theorem sequence:
kite_area_formula_diagonal(1,DABC)
Problem: As shown in the diagram, ∠CDO=40°, ∠OBA=20°, AB∥CD. Find the measure of ∠BOD.
问题：如图所示，∠CDO=40°，∠OBA=20°，AB∥CD。求∠BOD的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OC,CD,DO)
Shape(CO,OA)
Shape(BO,OD)
Collinear(AOD)
Collinear(BOC)
Conditions (CDL):
Equal(MeasureOfAngle(CDO),40)
Equal(MeasureOfAngle(OBA),20)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BOD))
Answer: 60
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
triangle_property_angle_sum(1,AOB)
adjacent_complementary_angle(1,AOB,BOD)
Problem: As shown in the diagram, ∠ABD=20°, the center of circle O is O, the diameter of ⊙O is DC. Find the measure of ∠CDA.
问题：如图所示，∠ABD=20°，O是圆O的圆心，DC是⊙O的直径。求∠CDA的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(EA,AC,CO,OE)
Shape(OCB,BE,EO,OC)
Shape(AE,ED,DA)
Shape(EB,BD,DE)
Shape(ODA,AD)
Shape(OBD,DB)
Collinear(COED)
Collinear(AEB)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DC,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,ODA,C)
diameter_of_circle_property_right_angle(1,DAC,O)
triangle_property_angle_sum(1,ACD)
Problem: As shown in the diagram, ∠APB=50°, O is the center of ⊙O, PA is the tangent to circle O, the tangent to circle O is PB. Find the measure of ∠BCA.
问题：如图所示，∠APB=50°，⊙O的圆心为O，圆O的切线为PA，⊙O的切线为PB。求∠BCA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(OCA,AC)
Shape(OA,OAB,BO)
Shape(AO,OB,BC,CA)
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, BC=DC, ∠COB=130°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，BC=DC，∠COB=130°，圆O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(DA,AB,BD)
Shape(DB,BO,OC,CD)
Shape(OB,BC,CO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(DC))
Equal(MeasureOfAngle(COB),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
isosceles_triangle_judgment_line_equal(1,CDB)
isosceles_triangle_property_angle_equal(1,CDB)
triangle_property_angle_sum(1,CDB)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, ∠ABO=15°, ∠CAB=80°, ∠OCA=40°. Find the measure of ∠COB.
问题：如图所示，∠ABO=15°，∠CAB=80°，∠OCA=40°。求∠COB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),15)
Equal(MeasureOfAngle(CAB),80)
Equal(MeasureOfAngle(OCA),40)
Goal: Value(MeasureOfAngle(COB))
Answer: 135
Theorem sequence:
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,OBC)
angle_addition(1,ABO,OBC)
angle_addition(1,BCO,OCA)
Problem: As shown in the diagram, AB=AC, ∠CAB=40°, ∠PBC=∠PCA. Find the measure of ∠CPB.
问题：如图所示，AB=AC，∠CAB=40°，∠PBC=∠PCA。求∠CPB的大小。
Construction (CDL):
Shape(CA,AB,BP,PC)
Shape(CP,PB,BC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),40)
Equal(MeasureOfAngle(PBC),MeasureOfAngle(PCA))
Goal: Value(MeasureOfAngle(CPB))
Answer: 110
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,PBC)
angle_addition(1,BCP,PCA)
Problem: As shown in the diagram, the area of quadrilateral ABCD is 25, the area of CFGE is 9, triangle DCE is a right triangle, quadrilateral ABCD is a square, quadrilateral CFGE is a square, quadrilateral DEHI is a square. Find the area of quadrilateral DEHI.
问题：如图所示，ABCD的面积为25，CFGE的面积为9，△DCE为直角△，四边形ABCD是正方形，四边形CFGE是正方形，DEHI是正方形。求DEHI的面积。
Construction (CDL):
Shape(AB,BC,CD,DA)
Shape(CF,FG,GE,EC)
Shape(DC,CE,ED)
Shape(DE,EH,HI,ID)
Collinear(DCF)
Collinear(BCE)
Conditions (CDL):
Equal(AreaOfQuadrilateral(ABCD),25)
Equal(AreaOfQuadrilateral(CFGE),9)
RightTriangle(DCE)
Square(ABCD)
Square(CFGE)
Square(DEHI)
Goal: Value(AreaOfQuadrilateral(DEHI))
Answer: 34
Theorem sequence:
right_triangle_property_pythagorean(1,DCE)
parallelogram_area_formula_sine(1,ABCD)
parallelogram_area_formula_sine(1,CFGE)
parallelogram_area_formula_sine(1,DEHI)
Problem: As shown in the diagram, AB=AC, BE=CD, ∠ABD=50°, ∠AEC=120°, triangle ABE is mirror congruent to triangle ADC. Find the measure of ∠CAD.
问题：如图所示，AB=AC，BE=CD，∠ABD=50°，∠AEC=120°，△ABE镜像全等于△ADC。求∠CAD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(AE,EC,CA)
Collinear(BDEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(BE),LengthOfLine(CD))
Equal(MeasureOfAngle(ABD),50)
Equal(MeasureOfAngle(AEC),120)
MirrorCongruentBetweenTriangle(ABE,ADC)
Goal: Value(MeasureOfAngle(CAD))
Answer: 70
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,BEA,CAD)
mirror_congruent_triangle_property_angle_equal(1,EAB,DCA)
adjacent_complementary_angle(1,DEA,AEC)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, AC=10, BC=7, DE perpendicular bisects AB. Find the perimeter of triangle BCD.
问题：如图所示，AC=10，BC=7，DE垂直平分AB。求三角形BCD的周长。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Shape(DB,BC,CD)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(BC),7)
IsPerpendicularBisectorOfLine(DE,AB)
Goal: Value(PerimeterOfTriangle(BCD))
Answer: 17
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AB)
line_addition(1,AD,DC)
triangle_perimeter_formula(1,BCD)
Problem: As shown in the diagram, ∠NOM=90°, AC is the angle bisector of ∠OAB, BE is the angle bisector of ∠NBA. Find the measure of ∠BCF.
问题：如图所示，∠NOM=90°，AC平分∠OAB，BE是∠NBA的角平分线。求∠BCF的大小。
Construction (CDL):
Shape(CB,BN)
Shape(NB,BE)
Shape(EB,BA)
Shape(BA,AM)
Shape(OF,FC)
Shape(CF,FB,BC)
Shape(BF,FA,AB)
Shape(FO,OA,AF)
Collinear(CBE)
Collinear(NBFO)
Collinear(OAM)
Collinear(CFA)
Conditions (CDL):
Equal(MeasureOfAngle(NOM),90)
IsBisectorOfAngle(AC,OAB)
IsBisectorOfAngle(BE,NBA)
Goal: Value(MeasureOfAngle(BCF))
Answer: 45
Theorem sequence:
triangle_property_angle_sum(1,BCA)
triangle_property_angle_sum(1,BOA)
adjacent_complementary_angle(1,EBA,ABC)
adjacent_complementary_angle(1,NBA,ABO)
angle_addition(1,NBE,EBA)
angle_addition(1,OAC,CAB)
Problem: As shown in the diagram, ∠ABC=30°, ∠DAC=25°, triangle ABC is mirror congruent to triangle ACD. Find the measure of ∠BCA.
问题：如图所示，∠ABC=30°，∠DAC=25°，△ABC与△ACD是镜像全等三角形。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
Equal(MeasureOfAngle(DAC),25)
MirrorCongruentBetweenTriangle(ABC,ACD)
Goal: Value(MeasureOfAngle(BCA))
Answer: 125
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,ABC,ACD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠DAB=34°, the center of circle O is O, AB is the tangent to ⊙O. Find the measure of ∠BCO.
问题：如图所示，∠DAB=34°，O是⊙O的圆心，AB是⊙O的切线。求∠BCO的大小。
Construction (CDL):
Shape(OCD,DO,OC)
Shape(DA,AB,ODB)
Shape(OD,ODB,BO)
Shape(OB,BC,CO)
Shape(OBC,CB)
Collinear(ADOC)
Cocircular(O,DBC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),34)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 28
Theorem sequence:
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,OAB)
adjacent_complementary_angle(1,COB,BOA)
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)
Problem: As shown in the diagram, AB=2, BD=15, OB=6, BA∥DC. Find the length of line CD.
问题：如图所示，AB=2，BD=15，OB=6，BA∥DC。求直线CD的长度。
Construction (CDL):
Shape(OB,BA,AO)
Shape(AB,BD,DC,CA)
Collinear(OBD)
Collinear(OAC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BD),15)
Equal(LengthOfLine(OB),6)
ParallelBetweenLine(BA,DC)
Goal: Value(LengthOfLine(CD))
Answer: 7
Theorem sequence:
parallel_property_corresponding_angle(1,BA,DC,O)
similar_triangle_judgment_aa(1,AOB,COD)
line_addition(1,OB,BD)
similar_triangle_property_line_ratio(1,AOB,COD)
similar_triangle_property_line_ratio(1,OBA,ODC)
Problem: As shown in the diagram, ∠CAB=50°, BD is the angle bisector of ∠ABC, CD bisects ∠BCA. Find the measure of ∠CDB.
问题：如图所示，∠CAB=50°，BD平分∠ABC，CD平分∠BCA。求∠CDB的大小。
Construction (CDL):
Shape(AB,BD,DC,CA)
Shape(DB,BC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsBisectorOfAngle(BD,ABC)
IsBisectorOfAngle(CD,BCA)
Goal: Value(MeasureOfAngle(CDB))
Answer: 115
Theorem sequence:
angle_addition(1,ABD,DBC)
angle_addition(1,BCD,DCA)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, AB=DC, ∠BCA=30°, ∠CAB=90°, ∠CDB=90°. Find the measure of ∠ACD.
问题：如图所示，AB=DC，∠BCA=30°，∠CAB=90°，∠CDB=90°。求∠ACD的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(DE,EC,CD)
Collinear(AEC)
Collinear(BED)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(DC))
Equal(MeasureOfAngle(BCA),30)
Equal(MeasureOfAngle(CAB),90)
Equal(MeasureOfAngle(CDB),90)
Goal: Value(MeasureOfAngle(ACD))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,CAB)
mirror_congruent_triangle_judgment_hl(1,CAB,BCD)
mirror_congruent_triangle_property_angle_equal(1,BCA,CDB)
angle_addition(1,BCA,ACD)
Problem: As shown in the diagram, AD=AE, BE=CD, ∠AEC=110°, ∠BDA=110°, ∠CAB=80°. Find the measure of ∠CAE.
问题：如图所示，AD=AE，BE=CD，∠AEC=110°，∠BDA=110°，∠CAB=80°。求∠CAE的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(AE,EC,CA)
Collinear(BDEC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(AE))
Equal(LengthOfLine(BE),LengthOfLine(CD))
Equal(MeasureOfAngle(AEC),110)
Equal(MeasureOfAngle(BDA),110)
Equal(MeasureOfAngle(CAB),80)
Goal: Value(MeasureOfAngle(CAE))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,BDA,ADE)
adjacent_complementary_angle(1,DEA,AEC)
mirror_congruent_triangle_judgment_sas(1,EAB,DCA)
mirror_congruent_triangle_property_angle_equal(1,BEA,CAD)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,AEC)
Problem: As shown in the diagram, AD=6, DB=15, DE=2, DE is parallel to BC. Find the length of line CB.
问题：如图所示，AD=6，DB=15，DE=2，DE平行于BC。求直线CB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(DB),15)
Equal(LengthOfLine(DE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(CB))
Answer: 7
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
line_addition(1,AD,DB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, ∠CAB=60°, ∠ENC=55°, DE is parallel to FG, triangle BCA is a right triangle. Find the measure of ∠FXM.
问题：如图所示，∠CAB=60°，∠ENC=55°，DE∥FG，∠BCA是直角三角形BCA的直角。求∠FXM的大小。
Construction (CDL):
Shape(DM,MA)
Shape(AN,NE)
Shape(EN,NC)
Shape(XM,MD)
Shape(CY,YG)
Shape(GY,YB)
Shape(BX,XF)
Shape(FX,XM)
Shape(XM,MD)
Shape(AM,MN,NA)
Shape(MX,XY,YC,CN,NM)
Shape(XB,BY,YX)
Collinear(DMNE)
Collinear(FXYG)
Collinear(AMXB)
Collinear(ANC)
Collinear(CYB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
Equal(MeasureOfAngle(ENC),55)
ParallelBetweenLine(DE,FG)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(FXM))
Answer: 115
Theorem sequence:
vertical_angle(1,MNA,ENC)
triangle_property_angle_sum(1,AMN)
vertical_angle(1,XMD,AMN)
parallel_property_collinear_extend(3,DE,FG,M)
parallel_property_collinear_extend(3,GF,MD,X)
parallel_property_ipsilateral_internal_angle(1,XF,MD)
Problem: As shown in the diagram, AD=2, AE=2, DB=1, DE∥BC. Find the length of line EC.
问题：如图所示，AD=2，AE=2，DB=1，DE平行于BC。求直线EC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(AE),2)
Equal(LengthOfLine(DB),1)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(EC))
Answer: 1
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
line_addition(1,AD,DB)
line_addition(1,AE,EC)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,DEA,BCA)
Problem: As shown in the diagram, AB=6, AC=BC, ∠BCA=90°, AD bisects ∠CAB, AE⊥DE. Find the perimeter of △DEB.
问题：如图所示，AB=6，AC=BC，∠BCA=90°，AD平分∠CAB，AE垂直于DE。求△DEB的周长。
Construction (CDL):
Shape(CA,AD,DC)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Collinear(CDB)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AC),LengthOfLine(BC))
Equal(MeasureOfAngle(BCA),90)
IsBisectorOfAngle(AD,CAB)
PerpendicularBetweenLine(AE,DE)
Goal: Value(PerimeterOfTriangle(DEB))
Answer: 6
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
bisector_of_angle_judgment_angle_equal(1,AD,CAE)
bisector_of_angle_property_distance_equal(1,AD,CAE)
isosceles_triangle_judgment_line_equal(1,CAB)
isosceles_triangle_property_angle_equal(1,CAB)
triangle_property_angle_sum(1,CAB)
adjacent_complementary_angle(1,AED,DEB)
triangle_property_angle_sum(1,DEB)
isosceles_triangle_judgment_angle_equal(1,EBD)
right_triangle_judgment_angle(1,DEB)
right_triangle_property_pythagorean(1,DEB)
line_addition(1,CD,DB)
triangle_perimeter_formula(1,DEB)
Problem: As shown in the diagram, ∠ABC=48°, AE is the angle bisector of ∠DAC, CE is the angle bisector of ∠ACF. Find the measure of ∠CEA.
问题：如图所示，∠ABC=48°，AE是∠DAC的角平分线，CE平分∠ACF。求∠CEA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DA,AE,ED)
Shape(AC,CE,EA)
Shape(EC,CF,FE)
Collinear(DAB)
Collinear(BCF)
Collinear(DEF)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),48)
IsBisectorOfAngle(AE,DAC)
IsBisectorOfAngle(CE,ACF)
Goal: Value(MeasureOfAngle(CEA))
Answer: 66
Theorem sequence:
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,ACE)
adjacent_complementary_angle(1,DAC,CAB)
adjacent_complementary_angle(1,BCA,ACF)
angle_addition(1,DAE,EAC)
angle_addition(1,ACE,ECF)
Problem: As shown in the diagram, AD=Mul(LengthOfLine(BC),2), ∠BCA=90°, ∠CDB=30°. Find the measure of ∠CAB.
问题：如图所示，AD=Mul(LengthOfLine(BC),2)，∠BCA=90°，∠CDB=30°。求∠CAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(CD,DB,BC)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AD),Mul(LengthOfLine(BC),2))
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CDB),30)
Goal: Value(MeasureOfAngle(CAB))
Answer: 15
Theorem sequence:
sine_theorem(1,BCD)
adjacent_complementary_angle(1,CDB,BDA)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,DAB)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠BAC=130°, AB∥CD, DE⊥CE. Find the measure of ∠CDE.
问题：如图所示，∠BAC=130°，AB∥CD，DE垂直于CE。求∠CDE的大小。
Construction (CDL):
Shape(AE,ED,DB,BA)
Shape(EC,CD,DE)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),130)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(DE,CE)
Goal: Value(MeasureOfAngle(CDE))
Answer: 40
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
triangle_property_angle_sum(1,ECD)
Problem: As shown in the diagram, ∠FEA=56°, EG is the angle bisector of ∠DEF, quadrilateral ABCD is a rectangle. Find the measure of ∠BGE.
问题：如图所示，∠FEA=56°，EG是∠DEF的角平分线，ABCD是长方形。求∠BGE的大小。
Construction (CDL):
Shape(AB,BF,FE,EA)
Shape(EF,FG,GE)
Shape(EG,GC,CD,DE)
Collinear(AED)
Collinear(BFGC)
Conditions (CDL):
Equal(MeasureOfAngle(FEA),56)
IsBisectorOfAngle(EG,DEF)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(BGE))
Answer: 62
Theorem sequence:
adjacent_complementary_angle(1,DEF,FEA)
angle_addition(1,DEG,GEF)
angle_addition(1,GEF,FEA)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,EA,G)
parallel_property_ipsilateral_internal_angle(1,GB,EA)
Problem: As shown in the diagram, ∠COB=110°, O is the center of ⊙O. Find the measure of ∠BDC.
问题：如图所示，∠COB=110°，⊙O的圆心为O。求∠BDC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BD,DC,CO)
Shape(DB,OBD)
Shape(CD,ODC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
quadrilateral_property_angle_sum(1,ABDC)
round_arc(1,OAD,ODA)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, AB=6/5, BP=9/5, PD=12, ∠BPA=∠CPD, AB is perpendicular to PB, PD⊥CD. Find the length of line CD.
问题：如图所示，AB=6/5，BP=9/5，PD=12，∠BPA=∠CPD，AB垂直于PB，PD⊥CD。求直线CD的长度。
Construction (CDL):
Shape(AB,BP,PA)
Shape(CP,PD,DC)
Collinear(BPD)
Conditions (CDL):
Equal(LengthOfLine(AB),6/5)
Equal(LengthOfLine(BP),9/5)
Equal(LengthOfLine(PD),12)
Equal(MeasureOfAngle(BPA),MeasureOfAngle(CPD))
PerpendicularBetweenLine(AB,PB)
PerpendicularBetweenLine(PD,CD)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,ABP,CPD)
mirror_similar_triangle_property_line_ratio(1,PAB,PDC)
mirror_similar_triangle_property_line_ratio(1,ABP,CPD)
Problem: As shown in the diagram, ∠COA=70°, O is the center of circle O, AD is parallel to CO. Find the measure of ∠AOD.
问题：如图所示，∠COA=70°，⊙O的圆心为O，AD∥CO。求∠AOD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OD)
Shape(CO,OA,OAC)
Shape(BO,OC,OCB)
Shape(DO,OB,OBD)
Collinear(AOB)
Cocircular(O,DACB)
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,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)
Problem: As shown in the diagram, ∠BAF=110°, ∠CEF=40°, AB∥FD. Find the measure of ∠FCE.
问题：如图所示，∠BAF=110°，∠CEF=40°，AB∥FD。求∠FCE的大小。
Construction (CDL):
Shape(BA,AF)
Shape(AF,FD)
Shape(DF,FE)
Shape(AF,FA)
Shape(FC,CE,EF)
Collinear(EFA)
Collinear(CFD)
Conditions (CDL):
Equal(MeasureOfAngle(BAF),110)
Equal(MeasureOfAngle(CEF),40)
ParallelBetweenLine(AB,FD)
Goal: Value(MeasureOfAngle(FCE))
Answer: 70
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,FD)
vertical_angle(1,AFD,EFC)
triangle_property_angle_sum(1,CEF)