SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BOD=66°, AD bisects ∠BAC, BE is the angle bisector of ∠CBA. Find the measure of ∠ACB.
问题：如图所示，∠BOD=66°，AD平分∠BAC，BE是∠CBA的角平分线。求∠ACB的大小。
Construction (CDL):
Shape(CD,DO,OE,EC)
Shape(OD,DB,BO)
Shape(EO,OA,AE)
Shape(AO,OB,BA)
Collinear(AEC)
Collinear(AOD)
Collinear(BOE)
Collinear(CDB)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),66)
IsBisectorOfAngle(AD,BAC)
IsBisectorOfAngle(BE,CBA)
Goal: Value(MeasureOfAngle(ACB))
Answer: 48
Theorem sequence:
adjacent_complementary_angle(1,AOB,BOD)
adjacent_complementary_angle(1,CDO,ODB)
triangle_property_angle_sum(1,ODB)
triangle_property_angle_sum(1,AOB)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠CBE=63°, ∠DAC=45°, AD∥BE, AD is parallel to FC, FC is parallel to BE. Find the measure of ∠BCA.
问题：如图所示，∠CBE=63°，∠DAC=45°，AD∥BE，AD平行于FC，FC∥BE。求∠BCA的大小。
Construction (CDL):
Shape(DA,AC)
Shape(FC,CA)
Shape(BC,CF)
Shape(CB,BE)
Conditions (CDL):
Equal(MeasureOfAngle(CBE),63)
Equal(MeasureOfAngle(DAC),45)
ParallelBetweenLine(AD,BE)
ParallelBetweenLine(AD,FC)
ParallelBetweenLine(FC,BE)
Goal: Value(MeasureOfAngle(BCA))
Answer: 108
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,FC)
parallel_property_alternate_interior_angle(2,FC,BE)
angle_addition(1,BCF,FCA)
Problem: As shown in the diagram, ∠ABD=26°, ∠DCA=90°, AD is the angle bisector of ∠CAB. Find the measure of ∠ADC.
问题：如图所示，∠ABD=26°，∠DCA=90°，AD平分∠CAB。求∠ADC的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AB,BD,DA)
Collinear(CDB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),26)
Equal(MeasureOfAngle(DCA),90)
IsBisectorOfAngle(AD,CAB)
Goal: Value(MeasureOfAngle(ADC))
Answer: 58
Theorem sequence:
adjacent_complementary_angle(1,BDA,ADC)
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, ∠CEM=80°, ∠MFC=50°, AB∥CF, CE is parallel to AD. Find the measure of ∠BAD.
问题：如图所示，∠CEM=80°，∠MFC=50°，AB∥CF，CE平行于AD。求∠BAD的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(AD,DC,CA)
Shape(CE,EM,MC)
Shape(CM,MF,FC)
Collinear(ACM)
Collinear(EMF)
Conditions (CDL):
Equal(MeasureOfAngle(CEM),80)
Equal(MeasureOfAngle(MFC),50)
ParallelBetweenLine(AB,CF)
ParallelBetweenLine(CE,AD)
Goal: Value(MeasureOfAngle(BAD))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,FCE)
parallel_property_corresponding_angle(1,CE,AD,M)
parallel_property_corresponding_angle(2,AB,CF,M)
angle_addition(1,BAC,CAD)
angle_addition(1,FCM,MCE)
Problem: As shown in the diagram, BC=6, BD=10, D is the midpoint of segment AC. Find the length of line AC.
问题：如图所示，BC=6，BD=10，D是线段AC的中点。求直线AC的长度。
Construction (CDL):
Shape(AD,DC,CB)
Collinear(ADCB)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(BD),10)
IsMidpointOfLine(D,AC)
Goal: Value(LengthOfLine(AC))
Answer: 8
Theorem sequence:
line_addition(1,AD,DC)
line_addition(1,DC,CB)
Problem: As shown in the diagram, ∠ACE=15°, ∠DBA=45°, BD is parallel to CE. Find the measure of ∠CAB.
问题：如图所示，∠ACE=15°，∠DBA=45°，BD平行于CE。求∠CAB的大小。
Construction (CDL):
Shape(DB,BA)
Shape(AB,BC,CA)
Shape(AC,CE)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),15)
Equal(MeasureOfAngle(DBA),45)
ParallelBetweenLine(BD,CE)
Goal: Value(MeasureOfAngle(CAB))
Answer: 60
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,BD,CE)
angle_addition(1,DBA,ABC)
angle_addition(1,BCA,ACE)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AO=6, the perimeter of quadrilateral EFGD is 14, BD is the median of triangle BCA, CE is the median of triangle CAB, F is the midpoint of segment OB, G is the midpoint of segment OC. Find the length of line BC.
问题：如图所示，AO=6，EFGD的周长为14，BD是三角形BCA的中线，CE是△CAB的中线，F是线段OB的中点，G是线段OC的中点。求直线BC的长度。
Construction (CDL):
Shape(AE,EH,HA)
Shape(AH,HD,DA)
Shape(EO,OH,HE)
Shape(HO,OD,DH)
Shape(EF,FO,OE)
Shape(OF,FG,GO)
Shape(OG,GD,DO)
Shape(EB,BF,FE)
Shape(FB,BC,CG,GF)
Shape(DG,GC,CD)
Collinear(AEB)
Collinear(AHO)
Collinear(ADC)
Collinear(EHD)
Collinear(BFOD)
Collinear(EOGC)
Conditions (CDL):
Equal(LengthOfLine(AO),6)
Equal(PerimeterOfQuadrilateral(EFGD),14)
IsMedianOfTriangle(BD,BCA)
IsMedianOfTriangle(CE,CAB)
IsMidpointOfLine(F,OB)
IsMidpointOfLine(G,OC)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
quadrilateral_perimeter_formula(1,EFGD)
midsegment_of_triangle_judgment_midpoint(1,FG,OBC)
midsegment_of_triangle_judgment_midpoint(1,FE,BOA)
midsegment_of_triangle_judgment_midpoint(1,DG,CAO)
midsegment_of_triangle_judgment_midpoint(1,ED,ABC)
midsegment_of_triangle_property_length(1,FG,OBC)
midsegment_of_triangle_property_length(1,FE,BOA)
midsegment_of_triangle_property_length(1,DG,CAO)
midsegment_of_triangle_property_length(1,ED,ABC)
Problem: As shown in the diagram, AB=8, ∠DAE=30°, D bisects segment AB, AC⊥BC, AE⊥DE. Find the length of line BC.
问题：如图所示，AB=8，∠DAE=30°，D是线段AB的中点，AC垂直于BC，AE⊥DE。求直线BC的长度。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EC,CD)
Shape(DC,CB,BD)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(DAE),30)
IsMidpointOfLine(D,AB)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(AE,DE)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
sine_theorem(1,BAC)
Problem: As shown in the diagram, BC=4, ∠ABC=90°, ∠CAB=30°, CF is the angle bisector of ∠ACM, DE is the midsegment of triangle ABC. Find the length of line DF.
问题：如图所示，BC=4，∠ABC=90°，∠CAB=30°，CF平分∠ACM，DE是三角形ABC的中位线。求直线DF的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Shape(EC,CF,FE)
Shape(FC,CM)
Collinear(ADB)
Collinear(AEC)
Collinear(DEF)
Collinear(BCM)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(MeasureOfAngle(ABC),90)
Equal(MeasureOfAngle(CAB),30)
IsBisectorOfAngle(CF,ACM)
IsMidsegmentOfTriangle(DE,ABC)
Goal: Value(LengthOfLine(DF))
Answer: 6
Theorem sequence:
sine_theorem(1,CAB)
line_addition(1,AE,EC)
midsegment_of_triangle_property_length(1,DE,ABC)
midsegment_of_triangle_property_parallel(1,DE,ABC)
parallel_property_collinear_extend(1,CB,ED,M)
parallel_property_collinear_extend(2,DE,CM,F)
parallel_property_alternate_interior_angle(2,DF,CM)
isosceles_triangle_judgment_angle_equal(1,ECF)
line_addition(1,DE,EF)
Problem: As shown in the diagram, BC=7, the perimeter of triangle ABC is 17, BD bisects ∠ABC, CE is the angle bisector of ∠BCA, AM⊥CM, BN is perpendicular to AN. Find the length of line MN.
问题：如图所示，BC=7，三角形ABC的周长为17，BD是∠ABC的角平分线，CE是∠BCA的角平分线，AM⊥CM，BN垂直于AN。求直线MN的长度。
Construction (CDL):
Shape(AE,EM,MA)
Shape(AM,MN,NA)
Shape(AN,ND,DA)
Shape(MO,ON,NM)
Shape(EB,BH,HM,ME)
Shape(MH,HO,OM)
Shape(HB,BG,GH)
Shape(OH,HG,GF,FI,IO)
Shape(OI,IN,NO)
Shape(NI,IC,CD,DN)
Shape(IF,FC,CI)
Collinear(EMOIC)
Collinear(BHOND)
Collinear(AEB)
Collinear(ADC)
Collinear(AMHG)
Collinear(ANIF)
Collinear(BGFC)
Conditions (CDL):
Equal(LengthOfLine(BC),7)
Equal(PerimeterOfTriangle(ABC),17)
IsBisectorOfAngle(BD,ABC)
IsBisectorOfAngle(CE,BCA)
PerpendicularBetweenLine(AM,CM)
PerpendicularBetweenLine(BN,AN)
Goal: Value(LengthOfLine(MN))
Answer: 3/2
Theorem sequence:
triangle_perimeter_formula(1,ABC)
adjacent_complementary_angle(1,FNB,BNA)
adjacent_complementary_angle(1,AMC,CMG)
triangle_property_angle_sum(1,ABN)
triangle_property_angle_sum(1,FNB)
triangle_property_angle_sum(1,AMC)
triangle_property_angle_sum(1,CMG)
isosceles_triangle_judgment_angle_equal(1,BFA)
isosceles_triangle_judgment_angle_equal(1,CAG)
altitude_of_triangle_judgment(1,BN,BFA)
altitude_of_triangle_judgment(1,CM,CAG)
isosceles_triangle_property_line_coincidence(1,BFA,N)
isosceles_triangle_property_line_coincidence(1,CAG,M)
midsegment_of_triangle_judgment_midpoint(1,MN,AGF)
midsegment_of_triangle_property_length(1,MN,AGF)
line_addition(1,BF,FC)
line_addition(1,GF,FC)
Problem: As shown in the diagram, AB=8, AC=5, BC=7, AD bisects ∠MAC, E is the midpoint of segment BC, CD⊥AD. Find the length of line DE.
问题：如图所示，AB=8，AC=5，BC=7，AD是∠MAC的角平分线，E平分线段BC，CD垂直于AD。求直线DE的长度。
Construction (CDL):
Shape(MA,AD,DM)
Shape(AB,BE,EO,OA)
Shape(OE,EC,CO)
Shape(AO,OD,DA)
Shape(OC,CD,DO)
Collinear(BAM)
Collinear(BEC)
Collinear(AOC)
Collinear(EOD)
Collinear(CDM)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(BC),7)
IsBisectorOfAngle(AD,MAC)
IsMidpointOfLine(E,BC)
PerpendicularBetweenLine(CD,AD)
Goal: Value(LengthOfLine(DE))
Answer: 13/2
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADM)
triangle_property_angle_sum(1,ACD)
triangle_property_angle_sum(1,ADM)
isosceles_triangle_judgment_angle_equal(1,ACM)
altitude_of_triangle_judgment(1,AD,ACM)
isosceles_triangle_property_line_coincidence(1,ACM,D)
line_addition(1,BA,AM)
midsegment_of_triangle_judgment_midpoint(1,DE,CMB)
midsegment_of_triangle_property_length(1,DE,CMB)
Problem: As shown in the diagram, AD=6, BC=18, E is the midpoint of segment BD, F is the midpoint of segment AC, AD∥BC. Find the length of line EF.
问题：如图所示，AD=6，BC=18，E平分线段BD，F是线段AC的中点，AD平行于BC。求直线EF的长度。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AB,BO,OA)
Shape(OE,EF,FO)
Shape(OF,FD,DO)
Shape(FC,CD,DF)
Shape(EB,BH,HF,FE)
Shape(FH,HC,CF)
Collinear(AOFC)
Collinear(BEOD)
Collinear(DFH)
Collinear(BHC)
Conditions (CDL):
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(BC),18)
IsMidpointOfLine(E,BD)
IsMidpointOfLine(F,AC)
ParallelBetweenLine(AD,BC)
Goal: Value(LengthOfLine(EF))
Answer: 6
Theorem sequence:
parallel_property_collinear_extend(3,CB,DA,H)
parallel_property_alternate_interior_angle(1,AD,BC)
parallel_property_alternate_interior_angle(2,CH,DA)
congruent_triangle_judgment_aas(3,FHC,FDA)
congruent_triangle_property_line_equal(1,FHC,FDA)
congruent_triangle_property_line_equal(1,CFH,AFD)
midsegment_of_triangle_judgment_midpoint(1,EF,DBH)
midsegment_of_triangle_property_length(1,EF,DBH)
line_addition(1,BH,HC)
Problem: As shown in the diagram, AC=4, BC=6, CD bisects ∠BCA, E is the midpoint of segment AB, AD⊥CD. Find the length of line DE.
问题：如图所示，AC=4，BC=6，CD是∠BCA的角平分线，E是线段AB的中点，AD⊥CD。求直线DE的长度。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BF,FD,DE)
Shape(AD,DC,CA)
Shape(DF,FC,CD)
Collinear(AEB)
Collinear(ADF)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),6)
IsBisectorOfAngle(CD,BCA)
IsMidpointOfLine(E,AB)
PerpendicularBetweenLine(AD,CD)
Goal: Value(LengthOfLine(DE))
Answer: 1
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDF)
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,DFC)
isosceles_triangle_judgment_angle_equal(1,CAF)
altitude_of_triangle_judgment(1,CD,CAF)
isosceles_triangle_property_line_coincidence(1,CAF,D)
line_addition(1,BF,FC)
midsegment_of_triangle_judgment_midpoint(1,ED,ABF)
midsegment_of_triangle_property_length(1,ED,ABF)
Problem: As shown in the diagram, EF=5, △ABC is an equilateral △, E bisects segment AB, F is the midpoint of segment AC. Find the perimeter of quadrilateral EBCF.
问题：如图所示，EF=5，三角形ABC为等边三角形，E平分线段AB，F是线段AC的中点。求四边形EBCF的周长。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BC,CF,FE)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(EF),5)
EquilateralTriangle(ABC)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,AC)
Goal: Value(PerimeterOfQuadrilateral(EBCF))
Answer: 25
Theorem sequence:
line_addition(1,AE,EB)
line_addition(1,AF,FC)
quadrilateral_perimeter_formula(1,EBCF)
midsegment_of_triangle_judgment_midpoint(1,EF,ABC)
midsegment_of_triangle_property_length(1,EF,ABC)
Problem: As shown in the diagram, AB=10, AC=16, AI is the angle bisector of ∠CAD, M bisects segment BC, AC is parallel to IM. Find the length of line MI.
问题：如图所示，AB=10，AC=16，AI平分∠CAD，M是线段BC的中点，AC∥IM。求直线MI的长度。
Construction (CDL):
Shape(AD,DI,IA)
Shape(AI,IM,MC,CA)
Shape(BM,MI,ID,DB)
Collinear(DIM)
Collinear(BMC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),16)
IsBisectorOfAngle(AI,CAD)
IsMidpointOfLine(M,BC)
ParallelBetweenLine(AC,IM)
Goal: Value(LengthOfLine(MI))
Answer: 3
Theorem sequence:
parallel_property_collinear_extend(2,MI,CA,D)
parallel_property_alternate_interior_angle(1,ID,CA)
isosceles_triangle_judgment_angle_equal(1,DIA)
midsegment_of_triangle_judgment_parallel(1,MD,BCA)
midsegment_of_triangle_property_length(1,MD,BCA)
line_addition(1,AD,DB)
line_addition(1,DI,IM)
Problem: As shown in the diagram, EF=2, OC=2, OD=3, EF is the midsegment of triangle ODB, AC is parallel to DB. Find the length of line AC.
问题：如图所示，EF=2，OC=2，OD=3，EF是三角形ODB的中位线，AC∥DB。求直线AC的长度。
Construction (CDL):
Shape(AO,OC,CA)
Shape(OE,EF,FO)
Shape(ED,DB,BF,FE)
Collinear(AOFB)
Collinear(COED)
Conditions (CDL):
Equal(LengthOfLine(EF),2)
Equal(LengthOfLine(OC),2)
Equal(LengthOfLine(OD),3)
IsMidsegmentOfTriangle(EF,ODB)
ParallelBetweenLine(AC,DB)
Goal: Value(LengthOfLine(AC))
Answer: 8/3
Theorem sequence:
midsegment_of_triangle_property_length(1,EF,ODB)
parallel_property_alternate_interior_angle(1,AC,DB)
parallel_property_alternate_interior_angle(2,AC,DB)
similar_triangle_judgment_aa(1,OCA,ODB)
similar_triangle_property_line_ratio(1,OCA,ODB)
similar_triangle_property_line_ratio(1,AOC,BOD)
Problem: As shown in the diagram, AB=6, BC=10, ∠CAB=90°, AD is the angle bisector of ∠CAB, AE is the median of triangle ABC, BG is perpendicular to AG, △CAB is a right △. Find the length of line EG.
问题：如图所示，AB=6，BC=10，∠CAB=90°，AD是∠CAB的角平分线，AE是三角形ABC的中线，BG垂直于AG，三角形CAB为直角三角形。求直线EG的长度。
Construction (CDL):
Shape(BG,GA,AB)
Shape(AG,GO,OA)
Shape(AO,OF,FA)
Shape(GB,BD,DG)
Shape(GD,DE,EG)
Shape(GE,EO,OG)
Shape(OE,EC,CF,FO)
Collinear(AFC)
Collinear(AOE)
Collinear(AGD)
Collinear(BGOF)
Collinear(BDEC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),10)
Equal(MeasureOfAngle(CAB),90)
IsBisectorOfAngle(AD,CAB)
IsMedianOfTriangle(AE,ABC)
PerpendicularBetweenLine(BG,AG)
RightTriangle(CAB)
Goal: Value(LengthOfLine(EG))
Answer: 1
Theorem sequence:
right_triangle_property_pythagorean(1,CAB)
adjacent_complementary_angle(1,BGA,AGF)
triangle_property_angle_sum(1,ABG)
triangle_property_angle_sum(1,AGF)
mirror_congruent_triangle_judgment_aas(1,GAB,GFA)
mirror_congruent_triangle_property_line_equal(1,GAB,GFA)
mirror_congruent_triangle_property_line_equal(1,ABG,AGF)
line_addition(1,AF,FC)
midsegment_of_triangle_judgment_midpoint(1,EG,BCF)
midsegment_of_triangle_property_length(1,EG,BCF)
Problem: As shown in the diagram, AB=CD, ∠MPN=140°, M is the midpoint of segment AD, N bisects segment BC, P bisects segment BD. Find the measure of ∠NMP.
问题：如图所示，AB=CD，∠MPN=140°，M平分线段AD，N是线段BC的中点，P是线段BD的中点。求∠NMP的大小。
Construction (CDL):
Shape(AB,BP,PM,MA)
Shape(MP,PO,OM)
Shape(MO,OD,DM)
Shape(PB,BN,NP)
Shape(PN,NO,OP)
Shape(DO,ON,NC,CD)
Collinear(AMD)
Collinear(BNC)
Collinear(BPOD)
Collinear(MON)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(MeasureOfAngle(MPN),140)
IsMidpointOfLine(M,AD)
IsMidpointOfLine(N,BC)
IsMidpointOfLine(P,BD)
Goal: Value(MeasureOfAngle(NMP))
Answer: 20
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,MP,DAB)
midsegment_of_triangle_judgment_midpoint(1,NP,BCD)
midsegment_of_triangle_property_length(1,MP,DAB)
midsegment_of_triangle_property_length(1,NP,BCD)
isosceles_triangle_judgment_line_equal(1,PNM)
isosceles_triangle_property_angle_equal(1,PNM)
triangle_property_angle_sum(1,MPN)
Problem: As shown in the diagram, AB=4, MN=3, AN bisects ∠BAD, M bisects segment BC, AN is perpendicular to BN. Find the length of line AC.
问题：如图所示，AB=4，MN=3，AN平分∠BAD，M平分线段BC，AN垂直于BN。求直线AC的长度。
Construction (CDL):
Shape(NB,BO,ON)
Shape(NA,AD,DN)
Shape(NO,OA,AN)
Shape(OB,BM,MO)
Shape(OM,MC,CA,AO)
Collinear(BOA)
Collinear(NOM)
Collinear(DNB)
Collinear(DAC)
Collinear(BMC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(MN),3)
IsBisectorOfAngle(AN,BAD)
IsMidpointOfLine(M,BC)
PerpendicularBetweenLine(AN,BN)
Goal: Value(LengthOfLine(AC))
Answer: 2
Theorem sequence:
adjacent_complementary_angle(1,DNA,ANB)
mirror_congruent_triangle_judgment_aas(2,DNA,BAN)
mirror_congruent_triangle_property_line_equal(1,NAD,NBA)
mirror_congruent_triangle_property_line_equal(1,ADN,ANB)
line_addition(1,DN,NB)
line_addition(1,DA,AC)
line_addition(1,BM,MC)
similar_triangle_judgment_sas(1,BMN,BCD)
similar_triangle_property_line_ratio(1,BMN,BCD)
similar_triangle_property_line_ratio(1,NBM,DBC)
Problem: As shown in the diagram, AB=4, AC=3, BC=2, BD bisects ∠ABC, CE is the angle bisector of ∠ACG, BD⊥AD, CE⊥AE, FD⊥BD. Find the length of line DE.
问题：如图所示，AB=4，AC=3，BC=2，BD是∠ABC的角平分线，CE平分∠ACG，BD垂直于AD，CE垂直于AE，FD⊥BD。求直线DE的长度。
Construction (CDL):
Shape(AB,BO,OA)
Shape(AO,OD,DA)
Shape(AD,DE,EA)
Shape(OB,BC,CO)
Shape(OC,CH,HD,DO)
Shape(DH,HE,ED)
Shape(HC,CF,FH)
Shape(EH,HF,FG,GE)
Collinear(AOC)
Collinear(ADHF)
Collinear(AEG)
Collinear(BOD)
Collinear(EHC)
Collinear(BCFG)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BC),2)
IsBisectorOfAngle(BD,ABC)
IsBisectorOfAngle(CE,ACG)
PerpendicularBetweenLine(BD,AD)
PerpendicularBetweenLine(CE,AE)
PerpendicularBetweenLine(FD,BD)
Goal: Value(LengthOfLine(DE))
Answer: 1/2
Theorem sequence:
mirror_congruent_triangle_judgment_aas(2,ABD,FDB)
line_addition(1,BC,CF)
adjacent_complementary_angle(1,GEC,CEA)
mirror_congruent_triangle_judgment_aas(2,ACE,GEC)
mirror_congruent_triangle_property_line_equal(1,BDA,BFD)
mirror_congruent_triangle_property_line_equal(1,DAB,DBF)
mirror_congruent_triangle_property_line_equal(1,CEA,CGE)
mirror_congruent_triangle_property_line_equal(1,EAC,ECG)
line_addition(1,CF,FG)
midsegment_of_triangle_judgment_midpoint(1,DE,AFG)
midsegment_of_triangle_property_length(1,DE,AFG)
Problem: As shown in the diagram, Div(LengthOfLine(DF)=LengthOfLine(EF)), BC=8, D is the midpoint of segment AB, E bisects segment AC. Find the length of line DF.
问题：如图所示，Div(LengthOfLine(DF)=LengthOfLine(EF))，BC=8，D是线段AB的中点，E是线段AC的中点。求直线DF的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(AE,EF,FA)
Shape(DB,BO,OE,ED)
Shape(EO,OF,FE)
Shape(OB,BC,CO)
Collinear(ADB)
Collinear(AEOC)
Collinear(DEF)
Collinear(BOF)
Conditions (CDL):
Equal(Div(LengthOfLine(DF),LengthOfLine(EF)),3)
Equal(LengthOfLine(BC),8)
IsMidpointOfLine(D,AB)
IsMidpointOfLine(E,AC)
Goal: Value(LengthOfLine(DF))
Answer: 6
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,DE,ABC)
midsegment_of_triangle_property_length(1,DE,ABC)
line_addition(1,DE,EF)
Problem: As shown in the diagram, AD=BC, ∠EPF=136°, E is the midpoint of segment AB, F is the midpoint of segment CD, P is the midpoint of segment DB. Find the measure of ∠PFE.
问题：如图所示，AD=BC，∠EPF=136°，E是线段AB的中点，F平分线段CD，P平分线段DB。求∠PFE的大小。
Construction (CDL):
Shape(AE,EO,OD,DA)
Shape(DO,OF,FD)
Shape(FO,OP,PF)
Shape(OE,EP,PO)
Shape(PE,EB,BP)
Shape(FP,PB,BC,CF)
Collinear(AEB)
Collinear(DOPB)
Collinear(FOE)
Collinear(DFC)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(BC))
Equal(MeasureOfAngle(EPF),136)
IsMidpointOfLine(E,AB)
IsMidpointOfLine(F,CD)
IsMidpointOfLine(P,DB)
Goal: Value(MeasureOfAngle(PFE))
Answer: 22
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,PE,BDA)
midsegment_of_triangle_judgment_midpoint(1,PF,DBC)
midsegment_of_triangle_property_length(1,PE,BDA)
midsegment_of_triangle_property_length(1,PF,DBC)
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, AB=5, BC=3, ∠BCA=90°. Find the length of line AC.
问题：如图所示，AB=5，BC=3，∠BCA=90°。求直线AC的长度。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(LengthOfLine(AC))
Answer: 4
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, AD=3, ∠BAC=105°, ∠CDA=90°, ∠EAB=45°, AE∥CB. Find the length of line AC.
问题：如图所示，AD=3，∠BAC=105°，∠CDA=90°，∠EAB=45°，AE∥CB。求直线AC的长度。
Construction (CDL):
Shape(CD,DA,AC)
Shape(AD,DB,BA)
Shape(AB,BE,EA)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(MeasureOfAngle(BAC),105)
Equal(MeasureOfAngle(CDA),90)
Equal(MeasureOfAngle(EAB),45)
ParallelBetweenLine(AE,CB)
Goal: Value(LengthOfLine(AC))
Answer: 6
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADB)
parallel_property_alternate_interior_angle(1,AE,CB)
triangle_property_angle_sum(1,ADB)
angle_addition(1,BAD,DAC)
triangle_property_angle_sum(1,ACD)
sine_theorem(1,ACD)
Problem: As shown in the diagram, OE=2, AO bisects ∠BAE, CO is the angle bisector of ∠ACD, AB∥CD, AE⊥OE, CD is perpendicular to OD, OB is perpendicular to AB. Find the length of line BD.
问题：如图所示，OE=2，AO平分∠BAE，CO是∠ACD的角平分线，AB∥CD，AE⊥OE，CD垂直于OD，OB⊥AB。求直线BD的长度。
Construction (CDL):
Shape(AO,OB,BA)
Shape(EO,OA,AE)
Shape(CO,OE,EC)
Shape(CD,DO,OC)
Collinear(AEC)
Collinear(BOD)
Conditions (CDL):
Equal(LengthOfLine(OE),2)
IsBisectorOfAngle(AO,BAE)
IsBisectorOfAngle(CO,ACD)
ParallelBetweenLine(AB,CD)
PerpendicularBetweenLine(AE,OE)
PerpendicularBetweenLine(CD,OD)
PerpendicularBetweenLine(OB,AB)
Goal: Value(LengthOfLine(BD))
Answer: 4
Theorem sequence:
adjacent_complementary_angle(1,AEO,OEC)
mirror_congruent_triangle_judgment_aas(1,OAE,OBA)
mirror_congruent_triangle_judgment_aas(1,OCD,OEC)
mirror_congruent_triangle_property_line_equal(1,AEO,AOB)
mirror_congruent_triangle_property_line_equal(1,CDO,COE)
line_addition(1,BO,OD)
Problem: As shown in the diagram, AB=3, AC=8. Find the length of line BC.
问题：如图所示，AB=3，AC=8。求直线BC的长度。
Construction (CDL):
Shape(AB,BC)
Collinear(ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AC),8)
Goal: Value(LengthOfLine(BC))
Answer: 5
Theorem sequence:
line_addition(1,AB,BC)
Problem: As shown in the diagram, AB=3, AD=4, AD is parallel to BC, AB⊥CB, CD is perpendicular to AD. Find the length of line BC.
问题：如图所示，AB=3，AD=4，AD平行于BC，AB⊥CB，CD⊥AD。求直线BC的长度。
Construction (CDL):
Shape(AB,BC,CD,DA)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),4)
ParallelBetweenLine(AD,BC)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(CD,AD)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
parallel_property_ipsilateral_internal_angle(1,CB,DA)
parallelogram_judgment_angle_and_angle(1,ABCD)
parallelogram_property_opposite_line_equal(1,BCDA)
Problem: As shown in the diagram, ∠BOE=40°, ∠COA=60°. Find the measure of ∠AOB.
问题：如图所示，∠BOE=40°，∠COA=60°。求∠AOB的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OD)
Shape(DO,OB)
Shape(BO,OE)
Collinear(COE)
Conditions (CDL):
Equal(MeasureOfAngle(BOE),40)
Equal(MeasureOfAngle(COA),60)
Goal: Value(MeasureOfAngle(AOB))
Answer: 80
Theorem sequence:
angle_addition(1,COA,AOB)
adjacent_complementary_angle(1,COB,BOE)
Problem: As shown in the diagram, AB=10, CD=3, D bisects segment AC. Find the length of line BC.
问题：如图所示，AB=10，CD=3，D平分线段AC。求直线BC的长度。
Construction (CDL):
Collinear(ADCB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(CD),3)
IsMidpointOfLine(D,AC)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
line_addition(1,AD,DC)
line_addition(1,AC,CB)
Problem: As shown in the diagram, AB=12, BC=Mul(LengthOfLine(AB),1/2), D is the midpoint of segment AC. Find the length of line BD.
问题：如图所示，AB=12，BC=Mul(LengthOfLine(AB),1/2)，D是线段AC的中点。求直线BD的长度。
Construction (CDL):
Collinear(ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(BC),Mul(LengthOfLine(AB),1/2))
IsMidpointOfLine(D,AC)
Goal: Value(LengthOfLine(BD))
Answer: 3
Theorem sequence:
line_addition(1,AB,BC)
line_addition(1,AD,DC)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠ACB=90°, ∠BAC=22°, △CBD is mirror congruent to △CDE. Find the measure of ∠DEC.
问题：如图所示，∠ACB=90°，∠BAC=22°，△CBD与△CDE是镜像全等三角形。求∠DEC的大小。
Construction (CDL):
Shape(CB,BD,DC)
Shape(CD,DE,EC)
Shape(ED,DA,AE)
Collinear(CEA)
Collinear(BDA)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(BAC),22)
MirrorCongruentBetweenTriangle(CBD,CDE)
Goal: Value(MeasureOfAngle(DEC))
Answer: 68
Theorem sequence:
triangle_property_angle_sum(1,ACB)
mirror_congruent_triangle_property_angle_equal(1,BDC,ECD)
Problem: As shown in the diagram, ∠AOC=50°, ∠BOE=25°, ∠DOC=90°, ∠EOD=90°. Find the measure of ∠BOA.
问题：如图所示，∠AOC=50°，∠BOE=25°，∠DOC=90°，∠EOD=90°。求∠BOA的大小。
Construction (CDL):
Shape(BO,OE)
Shape(EO,OD)
Shape(DO,OA)
Shape(AO,OC)
Collinear(EOC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),50)
Equal(MeasureOfAngle(BOE),25)
Equal(MeasureOfAngle(DOC),90)
Equal(MeasureOfAngle(EOD),90)
Goal: Value(MeasureOfAngle(BOA))
Answer: 155
Theorem sequence:
adjacent_complementary_angle(1,EOA,AOC)
angle_addition(1,BOE,EOA)