SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠NOE=20°, OE bisects ∠NOB, BO⊥AO. Find the measure of ∠AOM.
问题：如图所示，∠NOE=20°，OE是∠NOB的角平分线，BO垂直于AO。求∠AOM的大小。
Construction (CDL):
Shape(NO,OE)
Shape(EO,OB)
Shape(BO,OA)
Shape(AO,OM)
Shape(MO,OC)
Shape(CO,ON)
Collinear(BOC)
Collinear(NOM)
Conditions (CDL):
Equal(MeasureOfAngle(NOE),20)
IsBisectorOfAngle(OE,NOB)
PerpendicularBetweenLine(BO,AO)
Goal: Value(MeasureOfAngle(AOM))
Answer: 50
Theorem sequence:
angle_addition(1,NOE,EOB)
angle_addition(1,NOB,BOA)
adjacent_complementary_angle(1,NOA,AOM)
Problem: As shown in the diagram, ∠BAC=120°, AB is parallel to CD, AB∥EF, CA is parallel to FD, CD∥EF. Find the measure of ∠EFD.
问题：如图所示，∠BAC=120°，AB∥CD，AB平行于EF，CA平行于FD，CD平行于EF。求∠EFD的大小。
Construction (CDL):
Shape(BA,AC)
Shape(AC,CD)
Shape(FD,DC)
Shape(EF,FD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),120)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(AB,EF)
ParallelBetweenLine(CA,FD)
ParallelBetweenLine(CD,EF)
Goal: Value(MeasureOfAngle(EFD))
Answer: 120
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AB,CD)
parallel_property_ipsilateral_internal_angle(1,FE,DC)
parallel_property_alternate_interior_angle(1,CA,FD)
Problem: As shown in the diagram, ∠COA=37°. Find the measure of ∠AOD.
问题：如图所示，∠COA=37°。求∠AOD的大小。
Construction (CDL):
Shape(CO,OA)
Shape(AO,OD)
Collinear(COD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),37)
Goal: Value(MeasureOfAngle(AOD))
Answer: 143
Theorem sequence:
adjacent_complementary_angle(1,COA,AOD)
Problem: As shown in the diagram, ∠ACD=40°, OD is the angle bisector of ∠COB, CD∥OB. Find the measure of ∠ODC.
问题：如图所示，∠ACD=40°，OD平分∠COB，CD∥OB。求∠ODC的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(CO,OD,DC)
Shape(DO,OB,BD)
Collinear(ACO)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),40)
IsBisectorOfAngle(OD,COB)
ParallelBetweenLine(CD,OB)
Goal: Value(MeasureOfAngle(ODC))
Answer: 20
Theorem sequence:
parallel_property_corresponding_angle(1,CD,OB,A)
angle_addition(1,COD,DOB)
parallel_property_alternate_interior_angle(2,CD,OB)
Problem: As shown in the diagram, AB=2, AD=4, the radius of ⊙C is 2, the radius of ⊙F is 2, C is the center of circle C, F is the center of ⊙F, E bisects segment AD, F is the midpoint of segment BC, ABCD is a rectangle. The area of quadrilateral ABFE minus the area of sector FEB and plus the area of sector CDF.
问题：如图所示，AB=2，AD=4，圆C的半径为2，圆F的半径为2，⊙C的圆心为C，F是圆F的圆心，E是线段AD的中点，F平分线段BC，ABCD是矩形。求四边形ABFE的面积减去扇形FEB的面积加上扇形CDF面积。
Construction (CDL):
Shape(AB,FEB,EA)
Shape(BF,FE,FEB)
Shape(DE,EF,CDF)
Shape(FC,CD,CDF)
Collinear(AED)
Collinear(BFC)
Cocircular(F,EB)
Cocircular(C,DF)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(AD),4)
Equal(RadiusOfCircle(C),2)
Equal(RadiusOfCircle(F),2)
IsCentreOfCircle(C,C)
IsCentreOfCircle(F,F)
IsMidpointOfLine(E,AD)
IsMidpointOfLine(F,BC)
Rectangle(ABCD)
Goal: Value(Add(Sub(AreaOfQuadrilateral(ABFE),AreaOfSector(FEB)),AreaOfSector(CDF)))
Answer: 4
Theorem sequence:
line_addition(1,AE,ED)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,EA,F)
radius_of_circle_property_length_equal(1,FE,F)
radius_of_circle_property_length_equal(1,FB,F)
parallelogram_judgment_parallel_and_equal(1,EABF)
parallelogram_area_formula_sine(1,EABF)
parallelogram_area_formula_sine(1,BFEA)
sector_area_formula(1,FEB)
sector_area_formula(1,CDF)
arc_property_center_angle(1,FEB,F)
arc_property_center_angle(1,CDF,C)
Problem: As shown in the diagram, AB=AC, AD=AE, ∠ADE=x°, ∠EDC=20°. Find the measure of ∠DAB.
问题：如图所示，AB=AC，AD=AE，∠ADE=x°，∠EDC=20°。求∠DAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(BDC)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AD),LengthOfLine(AE))
Equal(MeasureOfAngle(ADE),x)
Equal(MeasureOfAngle(EDC),20)
Goal: Value(MeasureOfAngle(DAB))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ADE)
isosceles_triangle_property_angle_equal(1,ADE)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,ADE)
triangle_property_angle_sum(1,EDC)
triangle_property_angle_sum(1,ADC)
adjacent_complementary_angle(1,BDA,ADC)
adjacent_complementary_angle(1,CED,DEA)
Problem: As shown in the diagram, ∠BAE=150°, ∠CBF=150°. Find the measure of ∠ACD.
问题：如图所示，∠BAE=150°，∠CBF=150°。求∠ACD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(CB,BF)
Shape(BA,AE)
Shape(AC,CD)
Collinear(ABF)
Collinear(BCD)
Collinear(CAE)
Conditions (CDL):
Equal(MeasureOfAngle(BAE),150)
Equal(MeasureOfAngle(CBF),150)
Goal: Value(MeasureOfAngle(ACD))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,ABC,CBF)
adjacent_complementary_angle(1,BCA,ACD)
adjacent_complementary_angle(1,CAB,BAE)
Problem: As shown in the diagram, ∠EBA=120°, ∠ECD=25°, AB∥CD, AB∥FE, FE∥CD. Find the measure of ∠CEB.
问题：如图所示，∠EBA=120°，∠ECD=25°，AB平行于CD，AB∥FE，FE平行于CD。求∠CEB的大小。
Construction (CDL):
Shape(CE,EF)
Shape(FE,EB)
Shape(EB,BA)
Shape(EC,CD)
Conditions (CDL):
Equal(MeasureOfAngle(EBA),120)
Equal(MeasureOfAngle(ECD),25)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(AB,FE)
ParallelBetweenLine(FE,CD)
Goal: Value(MeasureOfAngle(CEB))
Answer: 85
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,EF,BA)
parallel_property_alternate_interior_angle(2,FE,CD)
angle_addition(1,CEF,FEB)
Problem: As shown in the diagram, ∠BCA=60°, ∠CAD=80°, CD is the angle bisector of ∠BCA. Find the measure of ∠CDB.
问题：如图所示，∠BCA=60°，∠CAD=80°，CD平分∠BCA。求∠CDB的大小。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DB,BC,CD)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),60)
Equal(MeasureOfAngle(CAD),80)
IsBisectorOfAngle(CD,BCA)
Goal: Value(MeasureOfAngle(CDB))
Answer: 110
Theorem sequence:
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,DBC)
triangle_property_angle_sum(1,CAB)
adjacent_complementary_angle(1,ADC,CDB)
Problem: As shown in the diagram, ∠BAD=60°, ∠CAB=90°, ∠CAE=60°. Find the measure of ∠EAH.
问题：如图所示，∠BAD=60°，∠CAB=90°，∠CAE=60°。求∠EAH的大小。
Construction (CDL):
Shape(CF,FE,EC)
Shape(CA,AF,FC)
Shape(FA,AH,HG,GF)
Shape(EF,FG,GE)
Shape(BG,GH,HB)
Shape(AD,DH,HA)
Collinear(CFGB)
Collinear(EFA)
Collinear(EGHD)
Collinear(AHB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),60)
Equal(MeasureOfAngle(CAB),90)
Equal(MeasureOfAngle(CAE),60)
Goal: Value(MeasureOfAngle(EAH))
Answer: 30
Theorem sequence:
angle_addition(1,CAE,EAH)
Problem: As shown in the diagram, quadrilateral BCDA is congruent to quadrilateral FGHE, HG=24, MC=6, MG=8, quadrilateral BCDA is a right trapezoid, quadrilateral FGHE is a right trapezoid. Find the area of quadrilateral MGHD.
问题：如图所示，四边形BCDA与四边形FGHE全等，HG=24，MC=6，MG=8，BC和DA是直角梯形BCDA的腰，FG和HE是直角梯形FGHE的腰。求四边形MGHD的面积。
Construction (CDL):
Shape(BC,CM,MF,FE,EA,AB)
Shape(FM,MD,DE,EF)
Shape(MG,GH,HD,DM)
Collinear(CMD)
Collinear(AEDH)
Collinear(FMG)
Conditions (CDL):
CongruentBetweenQuadrilateral(BCDA,FGHE)
Equal(LengthOfLine(HG),24)
Equal(LengthOfLine(MC),6)
Equal(LengthOfLine(MG),8)
RightTrapezoid(BCDA)
RightTrapezoid(FGHE)
Goal: Value(AreaOfQuadrilateral(MGHD))
Answer: 168
Theorem sequence:
congruent_quadrilateral_property_angle_equal(1,DABC,HEFG)
adjacent_complementary_angle(1,HDM,MDE)
parallel_judgment_ipsilateral_internal_angle(1,HG,DM)
trapezoid_judgment_parallel(1,MGHD)
right_trapezoid_judgment_right_angle(1,MGHD)
line_addition(1,CM,MD)
congruent_quadrilateral_property_line_equal(1,CDAB,GHEF)
right_trapezoid_area_formular(1,MGHD)
Problem: As shown in the diagram, AB=8, MN=3, AN bisects ∠DAB, M bisects segment BC, BN is perpendicular to AN. Find the length of line AC.
问题：如图所示，AB=8，MN=3，AN平分∠DAB，M是线段BC的中点，BN⊥AN。求直线AC的长度。
Construction (CDL):
Shape(AB,BN,NA)
Shape(NB,BM,MN)
Shape(AN,ND,DA)
Shape(NM,MC,CD,DN)
Collinear(ADC)
Collinear(BMC)
Collinear(BND)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(MN),3)
IsBisectorOfAngle(AN,DAB)
IsMidpointOfLine(M,BC)
PerpendicularBetweenLine(BN,AN)
Goal: Value(LengthOfLine(AC))
Answer: 14
Theorem sequence:
adjacent_complementary_angle(1,BNA,AND)
mirror_congruent_triangle_judgment_aas(2,BNA,DAN)
mirror_congruent_triangle_property_line_equal(1,NAB,NDA)
mirror_congruent_triangle_property_line_equal(1,ABN,AND)
midsegment_of_triangle_judgment_midpoint(1,MN,BCD)
midsegment_of_triangle_property_length(1,MN,BCD)
line_addition(1,AD,DC)
Problem: As shown in the diagram, Add(MeasureOfAngle(GCB)=MeasureOfAngle(FBA)), ∠HGQ=115°, AD∥EH. Find the measure of ∠EFB.
问题：如图所示，Add(MeasureOfAngle(GCB)=MeasureOfAngle(FBA))，∠HGQ=115°，AD∥EH。求∠EFB的大小。
Construction (CDL):
Shape(BF,FG,GC,CB)
Shape(AB,BM)
Shape(MB,BC)
Shape(BC,CR)
Shape(RC,CD)
Shape(DC,CG)
Shape(CG,GH)
Shape(HG,GQ)
Shape(QG,GF)
Shape(GF,FN)
Shape(NF,FE)
Shape(EF,FB)
Shape(FB,BA)
Collinear(ABCD)
Collinear(EFGH)
Collinear(MBFN)
Collinear(RCGQ)
Conditions (CDL):
Equal(Add(MeasureOfAngle(GCB),MeasureOfAngle(FBA)),90)
Equal(MeasureOfAngle(HGQ),115)
ParallelBetweenLine(AD,EH)
Goal: Value(MeasureOfAngle(EFB))
Answer: 155
Theorem sequence:
parallel_property_collinear_extend(3,AD,EH,B)
parallel_property_collinear_extend(3,AD,EH,C)
parallel_property_collinear_extend(3,HE,BA,F)
parallel_property_collinear_extend(3,HE,CA,G)
vertical_angle(1,HGQ,FGC)
parallel_property_ipsilateral_internal_angle(1,FE,BA)
parallel_property_ipsilateral_internal_angle(1,GE,CA)
Problem: As shown in the diagram, AB=AD, AD=DC, ∠ABD=72°. Find the measure of ∠CAD.
问题：如图所示，AB=AD，AD=DC，∠ABD=72°。求∠CAD的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(LengthOfLine(AD),LengthOfLine(DC))
Equal(MeasureOfAngle(ABD),72)
Goal: Value(MeasureOfAngle(CAD))
Answer: 36
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,DCA)
adjacent_complementary_angle(1,BDA,ADC)
triangle_property_angle_sum(1,ADC)
Problem: As shown in the diagram, ∠CAB=40°, DE is the perpendicular bisector of AC, AB and BC are the legs of the right △ ABC. Find the measure of ∠BCD.
问题：如图所示，∠CAB=40°，DE垂直平分AB，AB和BC是直角△ABC的两个直角边。求∠BCD的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Shape(DB,BC,CD)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsPerpendicularBisectorOfLine(DE,AC)
RightTriangle(ABC)
Goal: Value(MeasureOfAngle(BCD))
Answer: 10
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AC)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
triangle_property_angle_sum(1,ABC)
angle_addition(1,BCD,DCE)
Problem: As shown in the diagram, BD=DE, ∠EBD=25°, MQ is parallel to NP, triangle ACB is a right triangle. Find the measure of ∠CAP.
问题：如图所示，BD=DE，∠EBD=25°，MQ∥NP，AC和CB是直角△ACB的两个直角边。求∠CAP的大小。
Construction (CDL):
Shape(BD,DE,EB)
Shape(DA,AC,CE,ED)
Shape(MD,DB)
Shape(BE,EQ)
Shape(AD,DM)
Shape(NA,AD)
Shape(QE,EC)
Shape(CA,AP)
Collinear(BDA)
Collinear(BEC)
Collinear(MDEQ)
Collinear(NAP)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(DE))
Equal(MeasureOfAngle(EBD),25)
ParallelBetweenLine(MQ,NP)
RightTriangle(ACB)
Goal: Value(MeasureOfAngle(CAP))
Answer: 65
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DEB)
isosceles_triangle_property_angle_equal(1,DEB)
triangle_property_angle_sum(1,BDE)
triangle_property_angle_sum(1,BAC)
adjacent_complementary_angle(1,MDB,BDE)
parallel_property_collinear_extend(3,MQ,NP,D)
parallel_property_collinear_extend(3,PN,DM,A)
parallel_property_corresponding_angle(2,AN,DM,B)
angle_addition(1,NAD,DAC)
adjacent_complementary_angle(1,NAC,CAP)
Problem: As shown in the diagram, ∠EAN=58°, AN is parallel to BM, BM is perpendicular to AM. Find the measure of ∠MAB.
问题：如图所示，∠EAN=58°，AN平行于BM，BM⊥AM。求∠MAB的大小。
Construction (CDL):
Shape(AB,BM,MA)
Shape(EA,AN)
Shape(NA,AM)
Collinear(BAE)
Conditions (CDL):
Equal(MeasureOfAngle(EAN),58)
ParallelBetweenLine(AN,BM)
PerpendicularBetweenLine(BM,AM)
Goal: Value(MeasureOfAngle(MAB))
Answer: 32
Theorem sequence:
parallel_property_corresponding_angle(1,AN,BM,E)
triangle_property_angle_sum(1,ABM)
Problem: As shown in the diagram, the length of arc ODA is equal to the length of arc OCD, ∠CAO=26°, O is the center of ⊙O. Find the measure of ∠ACD.
问题：如图所示，⌒ODA的长度与⌒OCD的长度相等，∠CAO=26°，圆O的圆心为O。求∠ACD的大小。
Construction (CDL):
Shape(AO,OB,OBC,CA)
Shape(ODA,AD)
Shape(DA,AC,CD)
Shape(OCD,DC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(LengthOfArc(ODA),LengthOfArc(OCD))
Equal(MeasureOfAngle(CAO),26)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 32
Theorem sequence:
arc_addition_measure(1,OCD,ODA)
arc_addition_measure(1,OBC,OCA)
flat_angle(1,AOB)
arc_property_center_angle(1,OBA,O)
congruent_arc_judgment_length_equal(1,OCD,ODA)
congruent_arc_property_measure_equal(1,OCD,ODA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,ODA,C)
Problem: As shown in the diagram, ∠BCO=40°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=40°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, AB=20, AB=DE, AC=15, BF=15, BF=EC, CF=10, ∠ABF=∠CED. Find the sum of the perimeters of triangle ABC and triangle DFE.
问题：如图所示，AB=20，AB=DE，AC=15，BF=15，BF=EC，CF=10，∠ABF=∠CED。求三角形ABC和三角形DFE的周长和。
Construction (CDL):
Shape(AB,BF,FM,MA)
Shape(MF,FC,CM)
Shape(DM,MC,CE,ED)
Collinear(BFCE)
Collinear(AMC)
Collinear(DMF)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(LengthOfLine(AB),LengthOfLine(DE))
Equal(LengthOfLine(AC),15)
Equal(LengthOfLine(BF),15)
Equal(LengthOfLine(BF),LengthOfLine(EC))
Equal(LengthOfLine(CF),10)
Equal(MeasureOfAngle(ABF),MeasureOfAngle(CED))
Goal: Value(Sub(Add(PerimeterOfTriangle(ABC),PerimeterOfTriangle(DFE)),LengthOfLine(FC)))
Answer: 110
Theorem sequence:
line_addition(1,FC,CE)
line_addition(1,BC,CE)
line_addition(1,BF,FE)
mirror_congruent_triangle_judgment_sas(1,BCA,EDF)
mirror_congruent_triangle_property_line_equal(1,BCA,EDF)
triangle_perimeter_formula(1,ABC)
triangle_perimeter_formula(1,DFE)
Problem: As shown in the diagram, AB=AC, ∠EAO=40°, the center of ⊙O is O. Find the measure of ⌒OBD.
问题：如图所示，AB=AC，∠EAO=40°，圆O的圆心为O。求⌒OBD的角度。
Construction (CDL):
Shape(AO,OB,BD,DA)
Shape(AD,ODE,EA)
Shape(AE,OEA)
Shape(ODE,DC,CE)
Shape(DB,OBD)
Shape(OAB,BO,OA)
Collinear(AOB)
Collinear(BDC)
Collinear(AEC)
Cocircular(O,ABDE)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(EAO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfArc(OBD))
Answer: 40
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BDA,O)
isosceles_triangle_judgment_line_equal(1,ABC)
altitude_of_triangle_judgment(1,AD,ABC)
isosceles_triangle_property_line_coincidence(1,ABC,D)
angle_addition(1,EAD,DAO)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, AB=CD, ∠ABP=20°, ∠CDQ=70°, M is the midpoint of segment DA, N is the midpoint of segment BC, P is the midpoint of segment DB. Find the measure of ∠QMP.
问题：如图所示，AB=CD，∠ABP=20°，∠CDQ=70°，M是线段DA的中点，N是线段BC的中点，P是线段DB的中点。求∠QMP的大小。
Construction (CDL):
Shape(AB,BP,PM,MA)
Shape(MP,PQ,QM)
Shape(MQ,QD,DM)
Shape(PB,BN,NP)
Shape(QP,PN,NQ)
Shape(DQ,QN,NC,CD)
Collinear(AMD)
Collinear(BPQD)
Collinear(BNC)
Collinear(MQN)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(MeasureOfAngle(ABP),20)
Equal(MeasureOfAngle(CDQ),70)
IsMidpointOfLine(M,DA)
IsMidpointOfLine(N,BC)
IsMidpointOfLine(P,DB)
Goal: Value(MeasureOfAngle(QMP))
Answer: 25
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,MP,DAB)
midsegment_of_triangle_judgment_midpoint(1,NP,BCD)
midsegment_of_triangle_property_parallel(1,MP,DAB)
midsegment_of_triangle_property_parallel(1,NP,BCD)
midsegment_of_triangle_property_length(1,MP,DAB)
midsegment_of_triangle_property_length(1,NP,BCD)
parallel_property_corresponding_angle(2,BA,PM,D)
parallel_property_ipsilateral_internal_angle(1,DC,PN)
isosceles_triangle_judgment_line_equal(1,PNM)
isosceles_triangle_property_angle_equal(1,PNM)
angle_addition(1,MPQ,QPN)
triangle_property_angle_sum(1,PNM)
Problem: As shown in the diagram, ∠EOA=72°, ∠GOB=10°, HO⊥GO. Find the measure of ∠AOB.
问题：如图所示，∠EOA=72°，∠GOB=10°，HO⊥GO。求∠AOB的大小。
Construction (CDL):
Shape(EO,OA,AE)
Shape(AO,OH,HA)
Shape(HO,OG,GH)
Shape(GO,OB,BG)
Shape(BO,OF,FB)
Shape(FO,OE,EF)
Collinear(FOH)
Collinear(EOG)
Conditions (CDL):
Equal(MeasureOfAngle(EOA),72)
Equal(MeasureOfAngle(GOB),10)
PerpendicularBetweenLine(HO,GO)
Goal: Value(MeasureOfAngle(AOB))
Answer: 118
Theorem sequence:
angle_addition(1,HOG,GOB)
angle_addition(1,EOA,AOH)
adjacent_complementary_angle(1,EOH,HOG)
angle_addition(1,AOH,HOB)
Problem: As shown in the diagram, AB=40, ∠CAB=36°, ∠CBN=72°. Find the length of line BC.
问题：如图所示，AB=40，∠CAB=36°，∠CBN=72°。求直线BC的长度。
Construction (CDL):
Shape(CA,AB,BC)
Shape(CB,BN)
Collinear(ABN)
Conditions (CDL):
Equal(LengthOfLine(AB),40)
Equal(MeasureOfAngle(CAB),36)
Equal(MeasureOfAngle(CBN),72)
Goal: Value(LengthOfLine(BC))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBN)
triangle_property_angle_sum(1,CAB)
sine_theorem(1,BCA)
Problem: As shown in the diagram, ∠DCA=56°, ∠EBD=44°, AD is the angle bisector of ∠CAB, DE∥CA. Find the measure of ∠EDA.
问题：如图所示，∠DCA=56°，∠EBD=44°，AD是∠CAB的角平分线，DE∥CA。求∠EDA的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BD,DE)
Shape(AD,DC,CA)
Collinear(AEB)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),56)
Equal(MeasureOfAngle(EBD),44)
IsBisectorOfAngle(AD,CAB)
ParallelBetweenLine(DE,CA)
Goal: Value(MeasureOfAngle(EDA))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,DAB)
triangle_property_angle_sum(1,BCA)
angle_addition(1,CAD,DAE)
angle_addition(1,BDE,EDA)
parallel_property_corresponding_angle(1,DE,CA,B)
Problem: As shown in the diagram, AE=3, the perimeter of △ABD is 13, DE is the perpendicular bisector of AC. Find the perimeter of triangle ABC.
问题：如图所示，AE=3，三角形ABD的周长为13，DE垂直平分AB。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(AE),3)
Equal(PerimeterOfTriangle(ABD),13)
IsPerpendicularBisectorOfLine(DE,AC)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 19
Theorem sequence:
perpendicular_bisector_property_distance_equal(1,DE,AC)
line_addition(1,BD,DC)
line_addition(1,AE,EC)
triangle_perimeter_formula(1,ABD)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, ∠CED=50°, triangle ABC is an equilateral triangle, AD is the altitude of △ ABC. Find the measure of ∠ABE.
问题：如图所示，∠CED=50°，三角形ABC为等边三角形，AD是△ABC的高。求∠ABE的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(EB,BD,DE)
Shape(ED,DC,CE)
Collinear(AED)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(CED),50)
EquilateralTriangle(ABC)
IsAltitudeOfTriangle(AD,ABC)
Goal: Value(MeasureOfAngle(ABE))
Answer: 20
Theorem sequence:
isosceles_triangle_property_line_coincidence(1,ABC,D)
equilateral_triangle_property_angle(1,ABC)
angle_addition(1,CAE,EAB)
perpendicular_bisector_judgment_per_and_mid(1,ED,BC)
perpendicular_bisector_property_bisector(1,ED,BC)
triangle_property_angle_sum(1,ABE)
adjacent_complementary_angle(1,DEB,BEA)
Problem: As shown in the diagram, ∠BCP=30°, BC⊥AC, quadrilateral EFDG is a rectangle. Find the measure of ∠MNA.
问题：如图所示，∠BCP=30°，BC垂直于AC，四边形EFDG是矩形。求∠MNA的大小。
Construction (CDL):
Shape(AM,MN,NA)
Shape(MP,PC,CN,NM)
Shape(NC,CD,DG,GN)
Shape(EF,FP,PM,ME)
Shape(PB,BC,CP)
Collinear(AMPB)
Collinear(ANC)
Collinear(EMNG)
Collinear(FPCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCP),30)
PerpendicularBetweenLine(BC,AC)
Rectangle(EFDG)
Goal: Value(MeasureOfAngle(MNA))
Answer: 60
Theorem sequence:
angle_addition(1,BCP,PCN)
parallel_property_collinear_extend(3,EG,FD,N)
parallel_property_collinear_extend(3,DF,NE,C)
parallel_property_corresponding_angle(2,CF,NE,A)
Problem: As shown in the diagram, ∠EDB=60°, AD⊥CD, FD is perpendicular to ED. Find the measure of ∠FDC.
问题：如图所示，∠EDB=60°，AD⊥CD，FD⊥ED。求∠FDC的大小。
Construction (CDL):
Shape(AD,DF)
Shape(FD,DC)
Shape(CD,DE)
Shape(ED,DB)
Collinear(ADB)
Conditions (CDL):
Equal(MeasureOfAngle(EDB),60)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(FD,ED)
Goal: Value(MeasureOfAngle(FDC))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDB)
angle_addition(1,FDC,CDE)
angle_addition(1,CDE,EDB)
Problem: As shown in the diagram, AD=2, BC=5, the center of ⊙O is O, AD is the tangent to ⊙O, AF is the tangent to circle O, BD is the tangent to circle O, BE is the tangent to ⊙O, the tangent to ⊙O is CE, CF is the tangent to circle O. Find the perimeter of △ABC.
问题：如图所示，AD=2，BC=5，O是⊙O的圆心，圆O的切线为AD，AF是⊙O的切线，圆O的切线为BD，⊙O的切线为BE，CE是⊙O的切线，圆O的切线为CF。求三角形ABC的周长。
Construction (CDL):
Shape(AD,OFD,FA)
Shape(O)
Shape(OFD,ODE,OEF)
Shape(DB,BE,ODE)
Shape(OEF,EC,CF)
Collinear(ADB)
Collinear(BEC)
Collinear(AFC)
Cocircular(O,DEF)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(BC),5)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AD,O)
IsTangentOfCircle(AF,O)
IsTangentOfCircle(BD,O)
IsTangentOfCircle(BE,O)
IsTangentOfCircle(CE,O)
IsTangentOfCircle(CF,O)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 14
Theorem sequence:
line_addition(1,BE,EC)
line_addition(1,AD,DB)
line_addition(1,AF,FC)
tangent_of_circle_property_length_equal(1,BD,BE,O)
tangent_of_circle_property_length_equal(1,CE,CF,O)
tangent_of_circle_property_length_equal(1,AD,AF,O)
triangle_perimeter_formula(1,ABC)
Problem: As shown in the diagram, AB=3, BC=4, AB is perpendicular to CB. Find the sum of the length of line AB and the length of line AC.
问题：如图所示，AB=3，BC=4，AB垂直于CB。求直线AB的长度与直线AC的长度之和。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BC),4)
PerpendicularBetweenLine(AB,CB)
Goal: Value(Add(LengthOfLine(AB),LengthOfLine(AC)))
Answer: 8
Theorem sequence:
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠BOA=120°, OC bisects ∠BOA, OD is the angle bisector of ∠BOC. Find the measure of ∠DOA.
问题：如图所示，∠BOA=120°，OC是∠BOA的角平分线，OD是∠BOC的角平分线。求∠DOA的大小。
Construction (CDL):
Shape(BO,OD)
Shape(DO,OC)
Shape(CO,OA)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),120)
IsBisectorOfAngle(OC,BOA)
IsBisectorOfAngle(OD,BOC)
Goal: Value(MeasureOfAngle(DOA))
Answer: 90
Theorem sequence:
angle_addition(1,BOD,DOC)
angle_addition(1,BOD,DOA)
angle_addition(1,DOC,COA)
Problem: As shown in the diagram, BD=120, DC=60, EC=50, BA∥EC. Find the length of line AB.
问题：如图所示，BD=120，DC=60，EC=50，BA∥EC。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(DE,EC,CD)
Collinear(ADE)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(BD),120)
Equal(LengthOfLine(DC),60)
Equal(LengthOfLine(EC),50)
ParallelBetweenLine(BA,EC)
Goal: Value(LengthOfLine(AB))
Answer: 100
Theorem sequence:
parallel_property_alternate_interior_angle(1,BA,EC)
parallel_property_alternate_interior_angle(2,BA,EC)
similar_triangle_judgment_aa(1,DAB,DEC)
similar_triangle_property_line_ratio(1,DAB,DEC)
similar_triangle_property_line_ratio(1,ABD,ECD)
Problem: As shown in the diagram, ∠BEA=80°, ∠EBC=120°. Find the measure of ∠EAB.
问题：如图所示，∠BEA=80°，∠EBC=120°。求∠EAB的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC)
Collinear(ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BEA),80)
Equal(MeasureOfAngle(EBC),120)
Goal: Value(MeasureOfAngle(EAB))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,ABE,EBC)
triangle_property_angle_sum(1,ABE)