SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠EFA=40°, ∠FAE=30°, MNPQ is a rectangle, BC and CA are the legs of the right △ BCA. Find the measure of ∠NGE.
问题：如图所示，∠EFA=40°，∠FAE=30°，四边形MNPQ是矩形，∠BCA是直角△BCA的直角。求∠NGE的大小。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EG,GC,CF,FE)
Shape(MN,NG,GE,EM)
Shape(FC,CP,PQ,QF)
Shape(GB,BC,CG)
Collinear(AEGB)
Collinear(AFC)
Collinear(MEFQ)
Collinear(NGCP)
Conditions (CDL):
Equal(MeasureOfAngle(EFA),40)
Equal(MeasureOfAngle(FAE),30)
Rectangle(MNPQ)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(NGE))
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,AEF)
vertical_angle(1,GEM,AEF)
parallel_property_collinear_extend(3,MQ,NP,E)
parallel_property_collinear_extend(3,PN,EM,G)
parallel_property_ipsilateral_internal_angle(1,GN,EM)
Problem: As shown in the diagram, ∠CEM=70°, triangle EDA is mirror congruent to triangle EAM, ABCD is a rectangle. Find the measure of ∠AED.
问题：如图所示，∠CEM=70°，三角形EDA镜像全等于三角形EAM，四边形ABCD是长方形。求∠AED的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EA,AM,ME)
Shape(EM,MA,AB,BC,CE)
Collinear(DEC)
Conditions (CDL):
Equal(MeasureOfAngle(CEM),70)
MirrorCongruentBetweenTriangle(EDA,EAM)
Rectangle(ABCD)
Goal: Value(MeasureOfAngle(AED))
Answer: 55
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,EDA,EAM)
angle_addition(1,MEA,AED)
adjacent_complementary_angle(1,CEM,MED)
Problem: As shown in the diagram, ∠ABE=30°. Find the measure of ∠EBC.
问题：如图所示，∠ABE=30°。求∠EBC的大小。
Construction (CDL):
Shape(AB,BE)
Shape(EB,BC)
Shape(CB,BF)
Shape(FB,BA)
Collinear(ABC)
Collinear(EBF)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),30)
Goal: Value(MeasureOfAngle(EBC))
Answer: 150
Theorem sequence:
adjacent_complementary_angle(1,ABE,EBC)
Problem: As shown in the diagram, △ADE is congruent to △ABC, ∠CAE=20°. Find the measure of ∠MEA.
问题：如图所示，△ADE与△ABC是全等三角形，∠CAE=20°。求∠MEA的大小。
Construction (CDL):
Shape(AD,DM,MA)
Shape(AM,ME,EA)
Shape(AE,EC,CA)
Shape(MB,BE,EM)
Collinear(DME)
Collinear(AMB)
Collinear(BEC)
Conditions (CDL):
CongruentBetweenTriangle(ADE,ABC)
Equal(MeasureOfAngle(CAE),20)
Goal: Value(MeasureOfAngle(MEA))
Answer: 80
Theorem sequence:
congruent_triangle_property_line_equal(1,DEA,BCA)
isosceles_triangle_judgment_line_equal(1,AEC)
isosceles_triangle_property_angle_equal(1,AEC)
triangle_property_angle_sum(1,AEC)
congruent_triangle_property_angle_equal(1,EAD,CAB)
Problem: As shown in the diagram, ∠DEF=125°, AB∥CD. Find the measure of ∠EBA.
问题：如图所示，∠DEF=125°，AB平行于CD。求∠EBA的大小。
Construction (CDL):
Shape(AC,CE,EB,BA)
Shape(EF,FD,DE)
Collinear(BEF)
Collinear(CED)
Conditions (CDL):
Equal(MeasureOfAngle(DEF),125)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(EBA))
Answer: 55
Theorem sequence:
parallel_property_collinear_extend(3,DC,BA,E)
vertical_angle(1,CEB,DEF)
parallel_property_ipsilateral_internal_angle(1,EC,BA)
Problem: As shown in the diagram, ∠CAB=35°, ∠DEC=90°, CD∥AB. Find the measure of ∠CDE.
问题：如图所示，∠CAB=35°，∠DEC=90°，CD平行于AB。求∠CDE的大小。
Construction (CDL):
Shape(EC,CD,DE)
Shape(CA,AB,BD,DC)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
Equal(MeasureOfAngle(DEC),90)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(CDE))
Answer: 55
Theorem sequence:
parallel_property_corresponding_angle(1,CD,AB,E)
triangle_property_angle_sum(1,CDE)
Problem: As shown in the diagram, Sub(MeasureOfAngle(FCB)=MeasureOfAngle(ACE)), BC and CA are the legs of the right triangle BCA, EF and FG are the legs of the right △ EFG. Find the measure of ∠ACE.
问题：如图所示，Sub(MeasureOfAngle(FCB)=MeasureOfAngle(ACE))，△BCA为直角△，EF和FG是直角三角形EFG的两个直角边。求∠ACE的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(EC,CF,FG,GE)
Shape(AC,CE)
Shape(FC,CB)
Collinear(ECF)
Conditions (CDL):
Equal(Sub(MeasureOfAngle(FCB),MeasureOfAngle(ACE)),20)
RightTriangle(BCA)
RightTriangle(EFG)
Goal: Value(MeasureOfAngle(ACE))
Answer: 35
Theorem sequence:
angle_addition(1,FCB,BCA)
adjacent_complementary_angle(1,FCA,ACE)
Problem: As shown in the diagram, ∠BOC=70°, OE bisects ∠AOD. Find the measure of ∠COE.
问题：如图所示，∠BOC=70°，OE平分∠AOD。求∠COE的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OE)
Shape(EO,OD)
Shape(DO,OB)
Shape(BO,OC)
Collinear(COD)
Collinear(AOB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),70)
IsBisectorOfAngle(OE,AOD)
Goal: Value(MeasureOfAngle(COE))
Answer: 145
Theorem sequence:
adjacent_complementary_angle(1,BOC,COA)
adjacent_complementary_angle(1,COA,AOD)
angle_addition(1,AOE,EOD)
angle_addition(1,COA,AOE)
Problem: As shown in the diagram, AC=AB, ∠MEC=30°, CM perpendicular bisects AE. Find the measure of ∠BAM.
问题：如图所示，AC=AB，∠MEC=30°，CM垂直平分AB。求∠BAM的大小。
Construction (CDL):
Shape(ME,EC,CM)
Shape(AM,MC,CA)
Shape(AC,CB,BA)
Collinear(AME)
Collinear(BCE)
Conditions (CDL):
Equal(LengthOfLine(AC),LengthOfLine(AB))
Equal(MeasureOfAngle(MEC),30)
IsPerpendicularBisectorOfLine(CM,AE)
Goal: Value(MeasureOfAngle(BAM))
Answer: 90
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,CM,AE)
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_judgment_line_equal(1,CAE)
isosceles_triangle_property_angle_equal(1,ACB)
isosceles_triangle_property_angle_equal(1,CAE)
adjacent_complementary_angle(1,ECA,ACB)
triangle_property_angle_sum(1,ECA)
triangle_property_angle_sum(1,EBA)
Problem: As shown in the diagram, ∠DAB=40°, ABCD is a rhombus. Find the measure of ∠BDA.
问题：如图所示，∠DAB=40°，四边形ABCD是菱形。求∠BDA的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),40)
Rhombus(ABCD)
Goal: Value(MeasureOfAngle(BDA))
Answer: 70
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, AB=20, EB=3, C is the midpoint of segment AB, E bisects segment DB. Find the length of line CD.
问题：如图所示，AB=20，EB=3，C平分线段AB，E平分线段DB。求直线CD的长度。
Construction (CDL):
Shape(AC)
Shape(CD)
Shape(DE)
Shape(EB)
Collinear(ACDEB)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(LengthOfLine(EB),3)
IsMidpointOfLine(C,AB)
IsMidpointOfLine(E,DB)
Goal: Value(LengthOfLine(CD))
Answer: 4
Theorem sequence:
line_addition(1,AC,CD)
line_addition(1,AC,CB)
line_addition(1,AD,DE)
line_addition(1,AE,EB)
Problem: As shown in the diagram, ∠FCD=100°, CD is parallel to AB. Find the measure of ∠AEC.
问题：如图所示，∠FCD=100°，CD平行于AB。求∠AEC的大小。
Construction (CDL):
Shape(CE,EB,BD,DC)
Shape(FC,CD)
Shape(AE,EC)
Collinear(AEB)
Collinear(FCE)
Conditions (CDL):
Equal(MeasureOfAngle(FCD),100)
ParallelBetweenLine(CD,AB)
Goal: Value(MeasureOfAngle(AEC))
Answer: 80
Theorem sequence:
adjacent_complementary_angle(1,FCD,DCE)
parallel_property_collinear_extend(3,BA,DC,E)
parallel_property_alternate_interior_angle(1,EA,DC)
Problem: As shown in the diagram, ∠BCA=80°, ∠DAE=30°, ED perpendicular bisects CA. Find the measure of ∠BCE.
问题：如图所示，∠BCA=80°，∠DAE=30°，ED垂直平分AB。求∠BCE的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(CD,DE,EC)
Shape(CE,EB,BC)
Collinear(ADC)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),80)
Equal(MeasureOfAngle(DAE),30)
IsPerpendicularBisectorOfLine(ED,CA)
Goal: Value(MeasureOfAngle(BCE))
Answer: 50
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,ED,CA)
isosceles_triangle_judgment_line_equal(1,ECA)
isosceles_triangle_property_angle_equal(1,ECA)
angle_addition(1,BCE,ECA)
Problem: As shown in the diagram, AC=10, BD=6, E bisects segment AD, F bisects segment AB, G bisects segment BC, H bisects segment CD, AO⊥BO, BO⊥CO, CO⊥DO, DO⊥AO. Find the area of EHGF.
问题：如图所示，AC=10，BD=6，E平分线段AD，F平分线段AB，G平分线段BC，H是线段CD的中点，AO⊥BO，BO⊥CO，CO⊥DO，DO垂直于AO。求EHGF的面积。
Construction (CDL):
Shape(DH,HQ,QD)
Shape(DQ,QE,ED)
Shape(HC,CM,MH)
Shape(HM,MO,OQ,QH)
Shape(QO,OP,PE,EQ)
Shape(EP,PA,AE)
Shape(MC,CG,GM)
Shape(MG,GN,NO,OM)
Shape(ON,NF,FP,PO)
Shape(PF,FA,AP)
Shape(NG,GB,BN)
Shape(NB,BF,FN)
Collinear(AED)
Collinear(DHC)
Collinear(CGB)
Collinear(BFA)
Collinear(HMG)
Collinear(GNF)
Collinear(FPE)
Collinear(EQH)
Collinear(APOMC)
Collinear(DQONB)
Conditions (CDL):
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(BD),6)
IsMidpointOfLine(E,AD)
IsMidpointOfLine(F,AB)
IsMidpointOfLine(G,BC)
IsMidpointOfLine(H,CD)
PerpendicularBetweenLine(AO,BO)
PerpendicularBetweenLine(BO,CO)
PerpendicularBetweenLine(CO,DO)
PerpendicularBetweenLine(DO,AO)
Goal: Value(AreaOfQuadrilateral(EHGF))
Answer: 15
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,GH,CBD)
midsegment_of_triangle_judgment_midpoint(1,HE,DCA)
midsegment_of_triangle_judgment_midpoint(1,EF,ADB)
midsegment_of_triangle_judgment_midpoint(1,FG,BAC)
midsegment_of_triangle_property_parallel(1,GH,CBD)
midsegment_of_triangle_property_parallel(1,HE,DCA)
midsegment_of_triangle_property_length(1,GH,CBD)
midsegment_of_triangle_property_length(1,HE,DCA)
midsegment_of_triangle_property_length(1,EF,ADB)
midsegment_of_triangle_property_length(1,FG,BAC)
parallelogram_judgment_equal_and_equal(1,HGFE)
parallel_property_collinear_extend(3,HE,CA,Q)
parallel_property_collinear_extend(3,AC,QH,O)
parallel_property_collinear_extend(3,OC,QH,M)
parallel_property_collinear_extend(3,GH,BD,M)
parallel_property_collinear_extend(3,DB,HM,O)
parallel_property_collinear_extend(3,DO,HM,Q)
parallelogram_judgment_parallel_and_parallel(1,HMOQ)
parallelogram_property_opposite_angle_equal(1,HMOQ)
parallelogram_area_formula_sine(1,EHGF)
Problem: As shown in the diagram, AB=CE, ∠BCA=45°, ∠ECA=20°, AD⊥CD. Find the measure of ∠ABD.
问题：如图所示，AB=CE，∠BCA=45°，∠ECA=20°，AD垂直于CD。求∠ABD的大小。
Construction (CDL):
Shape(AB,BD,DE,EA)
Shape(AE,EC,CA)
Shape(CE,ED,DC)
Collinear(AED)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CE))
Equal(MeasureOfAngle(BCA),45)
Equal(MeasureOfAngle(ECA),20)
PerpendicularBetweenLine(AD,CD)
Goal: Value(MeasureOfAngle(ABD))
Answer: 65
Theorem sequence:
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,CED)
isosceles_triangle_judgment_angle_equal(1,DCA)
angle_addition(1,DCE,ECA)
adjacent_complementary_angle(1,BDE,EDC)
congruent_triangle_judgment_hl(2,BDA,EDC)
congruent_triangle_property_angle_equal(1,BDA,EDC)
Problem: As shown in the diagram, ∠ADE=35°, ∠BCA=120°, DE is parallel to BC. Find the measure of ∠EAD.
问题：如图所示，∠ADE=35°，∠BCA=120°，DE∥BC。求∠EAD的大小。
Construction (CDL):
Shape(EA,AD,DE)
Shape(CE,ED,DB,BC)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),35)
Equal(MeasureOfAngle(BCA),120)
ParallelBetweenLine(DE,BC)
Goal: Value(MeasureOfAngle(EAD))
Answer: 25
Theorem sequence:
parallel_property_corresponding_angle(2,CB,ED,A)
triangle_property_angle_sum(1,ADE)
Problem: As shown in the diagram, ∠ACE=20°, ∠BFA=40°, FE is perpendicular to CE. Find the measure of ∠ABF.
问题：如图所示，∠ACE=20°，∠BFA=40°，FE⊥CE。求∠ABF的大小。
Construction (CDL):
Shape(FE,ED,DF)
Shape(EA,AB,BD,DE)
Shape(DB,BC,CD)
Collinear(AEF)
Collinear(ABC)
Collinear(BDF)
Collinear(CDE)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),20)
Equal(MeasureOfAngle(BFA),40)
PerpendicularBetweenLine(FE,CE)
Goal: Value(MeasureOfAngle(ABF))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,FED,DEA)
triangle_property_angle_sum(1,FAB)
triangle_property_angle_sum(1,EAC)
Problem: As shown in the diagram, AB=AC, ∠CAB=50°, the center of ⊙O is O. Find the measure of ∠CDA.
问题：如图所示，AB=AC，∠CAB=50°，圆O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BC,CA)
Shape(O)
Shape(AC,CD,DA)
Shape(AD,ODA)
Shape(CB,OBC)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 115
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABC)
arc_addition_measure(1,OAB,OBC)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, the length of ⌒ACB is 50*pi, ∠BAC=150°, the center of ⊙A is A. Find the length of line AB.
问题：如图所示，弧ACB的长度为50*pi，∠BAC=150°，圆A的圆心为A。求直线AB的长度。
Construction (CDL):
Shape(BA,AC,ACB)
Shape(CA,AB,ABC)
Cocircular(A,BC)
Conditions (CDL):
Equal(LengthOfArc(ACB),50*pi)
Equal(MeasureOfAngle(BAC),150)
IsCentreOfCircle(A,A)
Goal: Value(LengthOfLine(AB))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,ACB,A)
arc_length_formula(1,ACB)
radius_of_circle_property_length_equal(1,AB,A)
Problem: As shown in the diagram, triangle ABC is congruent to triangle ADE, ∠DAB=110°, AE∥BD. Find the measure of ∠DAC.
问题：如图所示，三角形ABC全等于三角形ADE，∠DAB=110°，AE∥BD。求∠DAC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CB,BD,DA)
Shape(AD,DE,EA)
Conditions (CDL):
CongruentBetweenTriangle(ABC,ADE)
Equal(MeasureOfAngle(DAB),110)
ParallelBetweenLine(AE,BD)
Goal: Value(MeasureOfAngle(DAC))
Answer: 75
Theorem sequence:
congruent_triangle_property_line_equal(1,CAB,EAD)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
triangle_property_angle_sum(1,ABD)
parallel_property_ipsilateral_internal_angle(1,AE,BD)
angle_addition(1,EAD,DAB)
angle_addition(1,DAC,CAB)
congruent_triangle_property_angle_equal(1,ABC,ADE)
Problem: As shown in the diagram, AB=10, AC=6, BC=8, ∠BCA=90°, the incenter of triangle ABC is O, AF is perpendicular to OF, BD⊥OD, OE is perpendicular to AE. Find the length of line OF.
问题：如图所示，AB=10，AC=6，BC=8，∠BCA=90°，三角形ABC的内切圆圆心为O，AF垂直于OF，BD垂直于OD，OE⊥AE。求直线OF的长度。
Construction (CDL):
Shape(AF,FO,OA)
Shape(AO,OE,EA)
Shape(FB,BO,OF)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(CE,EO,OC)
Collinear(AFB)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(BCA),90)
IsIncenterOfTriangle(O,ABC)
PerpendicularBetweenLine(AF,OF)
PerpendicularBetweenLine(BD,OD)
PerpendicularBetweenLine(OE,AE)
Goal: Value(LengthOfLine(OF))
Answer: 2
Theorem sequence:
bisector_of_angle_judgment_angle_equal(1,AO,EAF)
bisector_of_angle_judgment_angle_equal(1,BO,FBD)
adjacent_complementary_angle(1,AFO,OFB)
adjacent_complementary_angle(1,CEO,OEA)
adjacent_complementary_angle(1,BDO,ODC)
bisector_of_angle_property_distance_equal(1,AO,EAF)
bisector_of_angle_property_distance_equal(1,BO,FBD)
mirror_congruent_triangle_judgment_hl(1,AFO,AOE)
mirror_congruent_triangle_judgment_hl(1,BDO,BOF)
mirror_congruent_triangle_property_line_equal(1,OAF,OEA)
mirror_congruent_triangle_property_line_equal(1,OBD,OFB)
quadrilateral_property_angle_sum(1,ODCE)
parallelogram_judgment_angle_and_angle(1,ODCE)
parallelogram_property_opposite_line_equal(1,ODCE)
parallelogram_property_opposite_line_equal(1,DCEO)
line_addition(1,AF,FB)
line_addition(1,BD,DC)
line_addition(1,AE,EC)
Problem: As shown in the diagram, ∠CAO=30°, ∠OCA=20°, triangle CAO is mirror congruent to triangle DOB. Find the measure of ∠DOC.
问题：如图所示，∠CAO=30°，∠OCA=20°，三角形CAO与三角形DOB是镜像全等三角形。求∠DOC的大小。
Construction (CDL):
Shape(AO,OE,EA)
Shape(DE,EG,GD)
Shape(GE,EO,OF,FG)
Shape(CG,GF,FC)
Shape(FO,OB,BF)
Collinear(AOB)
Collinear(DEO)
Collinear(CFO)
Collinear(DGFB)
Collinear(AEGC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),30)
Equal(MeasureOfAngle(OCA),20)
MirrorCongruentBetweenTriangle(CAO,DOB)
Goal: Value(MeasureOfAngle(DOC))
Answer: 80
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,OCA,OBD)
triangle_property_angle_sum(1,AOC)
adjacent_complementary_angle(1,AOD,DOB)
angle_addition(1,AOD,DOC)
Problem: As shown in the diagram, triangle ABC is congruent to triangle ADE, ∠ABC=40°, ∠EAC=60°. Find the measure of ∠CAD.
问题：如图所示，三角形ABC全等于三角形ADE，∠ABC=40°，∠EAC=60°。求∠CAD的大小。
Construction (CDL):
Shape(AB,BM,MA)
Shape(AM,MC,CA)
Shape(MD,DC,CM)
Shape(AC,CE,EA)
Collinear(AMD)
Collinear(DCE)
Collinear(BMC)
Conditions (CDL):
CongruentBetweenTriangle(ABC,ADE)
Equal(MeasureOfAngle(ABC),40)
Equal(MeasureOfAngle(EAC),60)
Goal: Value(MeasureOfAngle(CAD))
Answer: 20
Theorem sequence:
congruent_triangle_property_line_equal(1,BCA,DEA)
congruent_triangle_property_angle_equal(1,BCA,DEA)
isosceles_triangle_judgment_line_equal(1,ACE)
isosceles_triangle_property_angle_equal(1,ACE)
triangle_property_angle_sum(1,ACE)
triangle_property_angle_sum(1,ADE)
angle_addition(1,EAC,CAD)
Problem: As shown in the diagram, the area of triangle ADG is 50, the area of triangle AED is 25, DE=DG, AD bisects ∠CAB, AF is perpendicular to DF, DH⊥AH. Find the area of triangle EFD.
问题：如图所示，△ADG的面积为50，三角形AED的面积为25，DE=DG，AD是∠CAB的角平分线，AF⊥DF，DH垂直于AH。求△EFD的面积。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EF,FD,DE)
Shape(FB,BD,DF)
Shape(AD,DH,HA)
Shape(HD,DG,GH)
Shape(GD,DC,CG)
Collinear(AEFB)
Collinear(AHGC)
Collinear(BDC)
Conditions (CDL):
Equal(AreaOfTriangle(ADG),50)
Equal(AreaOfTriangle(AED),25)
Equal(LengthOfLine(DE),LengthOfLine(DG))
IsBisectorOfAngle(AD,CAB)
PerpendicularBetweenLine(AF,DF)
PerpendicularBetweenLine(DH,AH)
Goal: Value(AreaOfTriangle(EFD))
Answer: 25/2
Theorem sequence:
bisector_of_angle_judgment_angle_equal(1,AD,HAF)
bisector_of_angle_property_distance_equal(1,AD,HAF)
adjacent_complementary_angle(1,GHD,DHA)
congruent_triangle_judgment_hl(2,EFD,GHD)
congruent_triangle_property_line_equal(1,DEF,DGH)
mirror_congruent_triangle_judgment_hl(1,AFD,ADH)
mirror_congruent_triangle_property_line_equal(1,DAF,DHA)
altitude_of_triangle_judgment(3,DF,DAE)
altitude_of_triangle_judgment(1,DH,DGA)
line_addition(1,AE,EF)
line_addition(1,AH,HG)
triangle_area_formula_common(1,DAE)
triangle_area_formula_sine(1,FDE)
triangle_area_formula_common(1,DGA)
Problem: As shown in the diagram, ∠DOA=35°. Find the measure of ∠AOB.
问题：如图所示，∠DOA=35°。求∠AOB的大小。
Construction (CDL):
Shape(AO,OB)
Shape(BO,OC)
Shape(CO,OD)
Shape(DO,OA)
Collinear(AOC)
Collinear(BOD)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),35)
Goal: Value(MeasureOfAngle(AOB))
Answer: 145
Theorem sequence:
adjacent_complementary_angle(1,DOA,AOB)
Problem: As shown in the diagram, ∠CAP=20°, ∠EMP=40°, quadrilateral DEFG is a rectangle, BC and CA are the legs of the right triangle BCA. Find the measure of ∠PCA.
问题：如图所示，∠CAP=20°，∠EMP=40°，四边形DEFG是长方形，BC和CA是直角△BCA的两个直角边。求∠PCA的大小。
Construction (CDL):
Shape(AP,PC,CA)
Shape(DP,PA)
Shape(PM,MN,NC,CP)
Shape(MB,BN,NM)
Shape(CN,NF,FG,GC)
Shape(DE,EM,MP,PD)
Collinear(APMB)
Collinear(CNB)
Collinear(DPCG)
Collinear(EMNF)
Conditions (CDL):
Equal(MeasureOfAngle(CAP),20)
Equal(MeasureOfAngle(EMP),40)
Rectangle(DEFG)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(PCA))
Answer: 20
Theorem sequence:
parallel_property_collinear_extend(3,DG,EF,P)
parallel_property_collinear_extend(3,FE,PD,M)
parallel_property_corresponding_angle(2,ME,PD,A)
adjacent_complementary_angle(1,DPA,APC)
triangle_property_angle_sum(1,APC)
Problem: As shown in the diagram, ∠EFB=50°, AB∥CD. Find the measure of ∠DGH.
问题：如图所示，∠EFB=50°，AB平行于CD。求∠DGH的大小。
Construction (CDL):
Shape(AF,FE)
Shape(EF,FB)
Shape(BF,FG)
Shape(FG,GD)
Shape(DG,GH)
Shape(HG,GC)
Shape(CG,GF)
Shape(GF,FA)
Collinear(AFB)
Collinear(CGD)
Collinear(EFGH)
Conditions (CDL):
Equal(MeasureOfAngle(EFB),50)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(DGH))
Answer: 130
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_collinear_extend(3,DC,BF,G)
parallel_property_corresponding_angle(1,FB,GD,E)
adjacent_complementary_angle(1,FGD,DGH)
Problem: As shown in the diagram, ∠COA=50°, the center of ⊙O is O, DM is perpendicular to BM. Find the measure of ∠ABD.
问题：如图所示，∠COA=50°，圆O的圆心为O，DM垂直于BM。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(OA,AM,MO)
Shape(MA,OAC,CM)
Shape(DO,OM,MB,BD)
Shape(BM,MC,OCB)
Shape(DB,OBD)
Collinear(DOMC)
Collinear(AMB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DM,BM)
Goal: Value(MeasureOfAngle(ABD))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,COA,AOD)
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠BOA=72°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=72°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AO,OM,MC)
Shape(OA,AM,MO)
Shape(MB,BC,CM)
Shape(MA,OAB,BM)
Shape(CB,OBC)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 36
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠AOB=72°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=72°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OA,AC,CB,BO)
Shape(CA,OAC)
Shape(BC,OCB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 36
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, AD=BC, ∠FEP=25°, E is the midpoint of segment AB, F is the midpoint of segment CD, P bisects segment BD. Find the measure of ∠EPF.
问题：如图所示，AD=BC，∠FEP=25°，E是线段AB的中点，F是线段CD的中点，P平分线段BD。求∠EPF的大小。
Construction (CDL):
Shape(FD,DM,MF)
Shape(FM,MP,PF)
Shape(CF,FP,PB,BC)
Shape(MD,DA,AE,EM)
Shape(ME,EP,PM)
Shape(PE,EB,BP)
Collinear(DFC)
Collinear(DMPB)
Collinear(AEB)
Collinear(EMF)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(BC))
Equal(MeasureOfAngle(FEP),25)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,CD)
IsMidpointOfLine(P,BD)
Goal: Value(MeasureOfAngle(EPF))
Answer: 130
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,PF,DBC)
midsegment_of_triangle_judgment_midpoint(1,PE,BDA)
midsegment_of_triangle_property_length(1,PF,DBC)
midsegment_of_triangle_property_length(1,PE,BDA)
isosceles_triangle_judgment_line_equal(1,PFE)
isosceles_triangle_property_angle_equal(1,PFE)
triangle_property_angle_sum(1,PFE)
Problem: As shown in the diagram, ∠CBA=46°, ∠FEC=160°, AB∥EF, EF is parallel to CD. Find the measure of ∠BCE.
问题：如图所示，∠CBA=46°，∠FEC=160°，AB∥EF，EF平行于CD。求∠BCE的大小。
Construction (CDL):
Shape(CB,BA)
Shape(BC,CE,EF,FB)
Shape(EC,CD,DF,FE)
Collinear(BFD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),46)
Equal(MeasureOfAngle(FEC),160)
ParallelBetweenLine(AB,EF)
ParallelBetweenLine(EF,CD)
Goal: Value(MeasureOfAngle(BCE))
Answer: 26
Theorem sequence:
parallel_judgment_par_par(1,AB,EF,CD)
parallel_property_alternate_interior_angle(2,AB,CD)
parallel_property_ipsilateral_internal_angle(1,EF,CD)
angle_addition(1,BCE,ECD)
Problem: As shown in the diagram, ∠BCD=120°, O is the center of ⊙O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=120°，⊙O的圆心为O。求∠DOB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AB,BO,OD)
Shape(DO,OB,BC,CD)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
round_arc(1,ODB,OBD)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠ABE=20°, ∠ABE=∠DEB, ∠EAB=75°. Find the measure of ∠CED.
问题：如图所示，∠ABE=20°，∠ABE=∠DEB，∠EAB=75°。求∠CED的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BD,DE)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),20)
Equal(MeasureOfAngle(ABE),MeasureOfAngle(DEB))
Equal(MeasureOfAngle(EAB),75)
Goal: Value(MeasureOfAngle(CED))
Answer: 75
Theorem sequence:
parallel_judgment_alternate_interior_angle(1,BA,DE)
parallel_property_corresponding_angle(1,ED,AB,C)
Problem: As shown in the diagram, AD=AC, BD=AD, ∠CAE=16°, AE is the median of △ ADC. Find the measure of ∠ABD.
问题：如图所示，AD=AC，BD=AD，∠CAE=16°，AE是△ADC的中线。求∠ABD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(AC))
Equal(LengthOfLine(BD),LengthOfLine(AD))
Equal(MeasureOfAngle(CAE),16)
IsMedianOfTriangle(AE,ADC)
Goal: Value(MeasureOfAngle(ABD))
Answer: 37
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ADC)
isosceles_triangle_property_angle_equal(1,ADC)
isosceles_triangle_judgment_line_equal(1,DAB)
isosceles_triangle_property_angle_equal(1,DAB)
angle_addition(1,CAE,EAD)
angle_addition(1,CAD,DAB)
isosceles_triangle_property_line_coincidence(2,ADC,E)
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ABC)