SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, the area of ABXC is 169, AX=2*x, BC=x, ABXC is a rhombus. Find the length of line AX.
问题：如图所示，ABXC的面积为169，AX=2*x，BC=x，ABXC是菱形。求直线AX的长度。
Construction (CDL):
Shape(AO,OC,CA)
Shape(CO,OX,XC)
Shape(XO,OB,BX)
Shape(OA,AB,BO)
Collinear(COB)
Collinear(AOX)
Conditions (CDL):
Equal(AreaOfQuadrilateral(ABXC),169)
Equal(LengthOfLine(AX),2*x)
Equal(LengthOfLine(BC),x)
Rhombus(ABXC)
Goal: Value(LengthOfLine(AX))
Answer: 26
Theorem sequence:
kite_area_formula_diagonal(1,ABXC)
Problem: As shown in the diagram, RQ=4, UT=16, S is the midpoint of segment TR, V is the midpoint of segment QU, QUTR is a trapezoid. Find the length of line VS.
问题：如图所示，RQ=4，UT=16，S平分线段TR，V是线段QU的中点，QR和UT是梯形QUTR的平行边。求直线VS的长度。
Construction (CDL):
Shape(VU,UT,TS,SV)
Shape(QV,VS,SR,RQ)
Collinear(QVU)
Collinear(TSR)
Conditions (CDL):
Equal(LengthOfLine(RQ),4)
Equal(LengthOfLine(UT),16)
IsMidpointOfLine(S,TR)
IsMidpointOfLine(V,QU)
Trapezoid(QUTR)
Goal: Value(LengthOfLine(VS))
Answer: 10
Theorem sequence:
midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)
midsegment_of_quadrilateral_property_length(1,VS,QUTR)
Problem: As shown in the diagram, RS=4*sqrt(3), ST=10*sqrt(3), RS and ST are the legs of the right triangle RST. Find the measure of ∠STR.
问题：如图所示，RS=4*sqrt(3)，ST=10*sqrt(3)，△RST为直角△。求∠STR的大小。
Construction (CDL):
Shape(ST,TR,RS)
Conditions (CDL):
Equal(LengthOfLine(RS),4*sqrt(3))
Equal(LengthOfLine(ST),10*sqrt(3))
RightTriangle(RST)
Goal: Value(MeasureOfAngle(STR))
Answer: 180*acos(5*sqrt(29)/29)/pi
Theorem sequence:
cosine_theorem(1,TRS)
right_triangle_property_pythagorean(1,RST)
Problem: As shown in the diagram, AC=3*x, AD=x, BF=x+2, DF=8. Find the value of x.
问题：如图所示，AC=3*x，AD=x，BF=x+2，DF=8。求x的值。
Construction (CDL):
Cocircular(O,AC)
Cocircular(O,BF)
Collinear(DAC)
Collinear(DFB)
Conditions (CDL):
Equal(LengthOfLine(AC),3*x)
Equal(LengthOfLine(AD),x)
Equal(LengthOfLine(BF),x+2)
Equal(LengthOfLine(DF),8)
Goal: Value(x)
Answer: 1+sqrt(21)
Theorem sequence:
line_addition(1,DA,AC)
line_addition(1,DF,FB)
circle_property_circular_power_segment_and_segment_line(1,DAC,DFB,O)
Problem: As shown in the diagram, AC=2, BA=x, CB=5, BA and AC are the legs of the right △ BAC. Find the value of x.
问题：如图所示，AC=2，BA=x，CB=5，△BAC为直角△。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(BA),x)
Equal(LengthOfLine(CB),5)
RightTriangle(BAC)
Goal: Value(x)
Answer: sqrt(21)
Theorem sequence:
right_triangle_property_pythagorean(1,BAC)
Problem: As shown in the diagram, BA=3, CB=5, triangle BAC is a right triangle. Find the length of line AC.
问题：如图所示，BA=3，CB=5，∠BAC是直角三角形BAC的直角。求直线AC的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(BA),3)
Equal(LengthOfLine(CB),5)
RightTriangle(BAC)
Goal: Value(LengthOfLine(AC))
Answer: 4
Theorem sequence:
right_triangle_property_pythagorean(1,BAC)
Problem: As shown in the diagram, the height of CBAD is 10, AB=12, AD is perpendicular to CD, BA is perpendicular to DA, CB is perpendicular to AB, DC is perpendicular to BC. Find the area of CBAD.
问题：如图所示，CBAD的高为10，AB=12，AD垂直于CD，BA⊥DA，CB垂直于AB，DC垂直于BC。求CBAD的面积。
Construction (CDL):
Shape(CB,BA,AD,DC)
Conditions (CDL):
Equal(HeightOfQuadrilateral(CBAD),10)
Equal(LengthOfLine(AB),12)
PerpendicularBetweenLine(AD,CD)
PerpendicularBetweenLine(BA,DA)
PerpendicularBetweenLine(CB,AB)
PerpendicularBetweenLine(DC,BC)
Goal: Value(AreaOfQuadrilateral(CBAD))
Answer: 120
Theorem sequence:
parallel_judgment_per_per(1,BC,AD)
parallel_judgment_per_per(1,AB,DC)
parallelogram_judgment_parallel_and_parallel(1,CBAD)
parallelogram_area_formula_common(1,CBAD)
Problem: As shown in the diagram, AC=14, AD=15, DB=10, BD is perpendicular to AD, DA is perpendicular to CA, DB and AC are the parallel sides of trapezoid DACB. Find the area of DACB.
问题：如图所示，AC=14，AD=15，DB=10，BD⊥AD，DA垂直于CA，四边形DACB是梯形。求四边形DACB的面积。
Construction (CDL):
Shape(BD,DA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AC),14)
Equal(LengthOfLine(AD),15)
Equal(LengthOfLine(DB),10)
PerpendicularBetweenLine(BD,AD)
PerpendicularBetweenLine(DA,CA)
Trapezoid(DACB)
Goal: Value(AreaOfQuadrilateral(DACB))
Answer: 180
Theorem sequence:
right_trapezoid_judgment_right_angle(1,DACB)
right_trapezoid_area_formular(1,DACB)
Problem: As shown in the diagram, the area of KLMJ is 121, the area of quadrilateral QRST is 11, KJ=5, QT=x, KLMJ is similar to QRST. Find the value of x.
问题：如图所示，四边形KLMJ的面积为121，QRST的面积为11，KJ=5，QT=x，四边形KLMJ相似于四边形QRST。求x的值。
Construction (CDL):
Shape(KL,LM,MJ,JK)
Shape(QR,RS,ST,TQ)
Conditions (CDL):
Equal(AreaOfQuadrilateral(KLMJ),121)
Equal(AreaOfQuadrilateral(QRST),11)
Equal(LengthOfLine(KJ),5)
Equal(LengthOfLine(QT),x)
SimilarBetweenQuadrilateral(KLMJ,QRST)
Goal: Value(x)
Answer: 5*sqrt(11)/11
Theorem sequence:
similar_quadrilateral_property_area_square_ratio(1,KLMJ,QRST)
similar_quadrilateral_property_line_ratio(1,JKLM,TQRS)
Problem: As shown in the diagram, AC=9, BA=6, BC=15/2, DE=9, BF is the angle bisector of ∠ABC, CF is parallel to DE. Find the length of line CF.
问题：如图所示，AC=9，BA=6，BC=15/2，DE=9，BF平分∠ABC，CF∥DE。求直线CF的长度。
Construction (CDL):
Shape(AB,BF,FA)
Shape(FB,BC,CF)
Shape(FC,CD,DE,EF)
Collinear(BFE)
Collinear(BCD)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(AC),9)
Equal(LengthOfLine(BA),6)
Equal(LengthOfLine(BC),15/2)
Equal(LengthOfLine(DE),9)
IsBisectorOfAngle(BF,ABC)
ParallelBetweenLine(CF,DE)
Goal: Value(LengthOfLine(CF))
Answer: 5
Theorem sequence:
bisector_of_angle_property_line_ratio(1,BF,ABC)
line_addition(1,AF,FC)
Problem: As shown in the diagram, ∠WXJ=130°, ∠YZI=20°, Mul(LengthOfLine(IJ)=LengthOfLine(YJ)). Find the measure of ∠YJX.
问题：如图所示，∠WXJ=130°，∠YZI=20°，Mul(LengthOfLine(IJ)=LengthOfLine(YJ))。求∠YJX的大小。
Construction (CDL):
Shape(JH,HI,IJ)
Shape(WX,XJ)
Shape(HX,XY,YI,IH)
Shape(IY,YZ,ZI)
Collinear(DWXYZC)
Collinear(JHX)
Collinear(YIJ)
Collinear(ZIHG)
Conditions (CDL):
Equal(MeasureOfAngle(WXJ),130)
Equal(MeasureOfAngle(YZI),20)
Equal(Mul(LengthOfLine(IJ),LengthOfLine(YJ)),Mul(LengthOfLine(JX),LengthOfLine(JH)))
Goal: Value(MeasureOfAngle(YJX))
Answer: 60
Theorem sequence:
mirror_similar_triangle_judgment_sas(1,JHI,JXY)
mirror_similar_triangle_property_angle_equal(1,IJH,XYJ)
vertical_angle(1,HIJ,ZIY)
adjacent_complementary_angle(1,WXJ,JXC)
triangle_property_angle_sum(1,IYZ)
adjacent_complementary_angle(1,XYJ,JYC)
triangle_property_angle_sum(1,JXY)
Problem: As shown in the diagram, JK=13, KL=5, ∠KLJ is the right angle of the right triangle KLJ. Find the measure of ∠LJK.
问题：如图所示，JK=13，KL=5，∠KLJ是直角三角形KLJ的直角。求∠LJK的大小。
Construction (CDL):
Shape(LJ,JK,KL)
Conditions (CDL):
Equal(LengthOfLine(JK),13)
Equal(LengthOfLine(KL),5)
RightTriangle(KLJ)
Goal: Value(MeasureOfAngle(LJK))
Answer: 180*acos(12/13)/pi
Theorem sequence:
right_triangle_property_pythagorean(1,KLJ)
cosine_theorem(1,JKL)
Problem: As shown in the diagram, FA=4*w-7, GA=2*b+5, HA=2*w+3, JA=3*b+1, FJHG is a parallelogram. Find the value of w.
问题：如图所示，FA=4*w-7，GA=2*b+5，HA=2*w+3，JA=3*b+1，四边形FJHG是平行四边形。求w的值。
Construction (CDL):
Shape(FJ,JA,AF)
Shape(JH,HA,AJ)
Shape(AH,HG,GA)
Shape(AG,GF,FA)
Collinear(JAG)
Collinear(FAH)
Conditions (CDL):
Equal(LengthOfLine(FA),4*w-7)
Equal(LengthOfLine(GA),2*b+5)
Equal(LengthOfLine(HA),2*w+3)
Equal(LengthOfLine(JA),3*b+1)
Parallelogram(FJHG)
Goal: Value(w)
Answer: 5
Theorem sequence:
parallelogram_property_diagonal_bisection(1,FJHG,A)
Problem: As shown in the diagram, AB=2*x, CD=14, ST=10, S is the midpoint of segment AC, T bisects segment BD, quadrilateral ACDB is a trapezoid. Find the value of x.
问题：如图所示，AB=2*x，CD=14，ST=10，S平分线段AC，T平分线段BD，ACDB是梯形。求x的值。
Construction (CDL):
Shape(SC,CD,DT,TS)
Shape(AS,ST,TB,BA)
Collinear(ASC)
Collinear(BTD)
Conditions (CDL):
Equal(LengthOfLine(AB),2*x)
Equal(LengthOfLine(CD),14)
Equal(LengthOfLine(ST),10)
IsMidpointOfLine(S,AC)
IsMidpointOfLine(T,BD)
Trapezoid(ACDB)
Goal: Value(x)
Answer: 3
Theorem sequence:
midsegment_of_quadrilateral_judgment_midpoint(1,ST,ACDB)
midsegment_of_quadrilateral_property_length(1,ST,ACDB)
Problem: As shown in the diagram, ∠CBF=37°, DB⊥CB. Find the measure of ∠GBD.
问题：如图所示，∠CBF=37°，DB⊥CB。求∠GBD的大小。
Construction (CDL):
Shape(AB,BG)
Shape(GB,BD)
Shape(DB,BC)
Shape(CB,BF)
Shape(FB,BE)
Shape(EB,BA)
Collinear(GBF)
Collinear(DBE)
Collinear(ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CBF),37)
PerpendicularBetweenLine(DB,CB)
Goal: Value(MeasureOfAngle(GBD))
Answer: 53
Theorem sequence:
vertical_angle(1,ABG,CBF)
flat_angle(1,ABC)
angle_addition(1,ABG,GBC)
angle_addition(1,GBD,DBC)
Problem: As shown in the diagram, AC=x, BA=14*sqrt(3), CB=y, ∠CBA=30°, BA and AC are the legs of the right triangle BAC. Find the value of y.
问题：如图所示，AC=x，BA=14*sqrt(3)，CB=y，∠CBA=30°，∠BAC是直角三角形BAC的直角。求y的值。
Construction (CDL):
Shape(CB,BA,AC)
Conditions (CDL):
Equal(LengthOfLine(AC),x)
Equal(LengthOfLine(BA),14*sqrt(3))
Equal(LengthOfLine(CB),y)
Equal(MeasureOfAngle(CBA),30)
RightTriangle(BAC)
Goal: Value(y)
Answer: 28
Theorem sequence:
triangle_property_angle_sum(1,ACB)
sine_theorem(1,BAC)
Problem: As shown in the diagram, AB=12, AE=2, CD=8, DA=CB, ∠ADE=30°, the height of CBAD is DE, DE⊥AE, CBAD is a trapezoid. Find the area of quadrilateral CBAD.
问题：如图所示，AB=12，AE=2，CD=8，DA=CB，∠ADE=30°，DE是四边形CBAD的高，DE⊥AE，四边形CBAD是梯形。求四边形CBAD的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DC,CB,BE)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(AE),2)
Equal(LengthOfLine(CD),8)
Equal(LengthOfLine(DA),LengthOfLine(CB))
Equal(MeasureOfAngle(ADE),30)
IsAltitudeOfQuadrilateral(DE,CBAD)
PerpendicularBetweenLine(DE,AE)
Trapezoid(CBAD)
Goal: Value(AreaOfQuadrilateral(CBAD))
Answer: 20*sqrt(3)
Theorem sequence:
cosine_theorem(1,DEA)
cosine_theorem(1,EAD)
trapezoid_area_formula(1,CBAD)
Problem: As shown in the diagram, MP=36, NM=39, PN=15, ∠MPN is the right angle of the right △ MPN. Find the value of tan(NM).
问题：如图所示，MP=36，NM=39，PN=15，∠MPN是直角三角形MPN的直角。求tan(NM)的值。
Construction (CDL):
Shape(PN,NM,MP)
Conditions (CDL):
Equal(LengthOfLine(MP),36)
Equal(LengthOfLine(NM),39)
Equal(LengthOfLine(PN),15)
RightTriangle(MPN)
Goal: Value(Tan(MeasureOfAngle(NMP)))
Answer: 5/12
Theorem sequence:
cosine_theorem(1,MPN)
Problem: As shown in the diagram, AC=9, BA=15, CB=x, AC and CB are the legs of the right △ ACB. Find the value of x.
问题：如图所示，AC=9，BA=15，CB=x，AC和CB是直角三角形ACB的两个直角边。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),9)
Equal(LengthOfLine(BA),15)
Equal(LengthOfLine(CB),x)
RightTriangle(ACB)
Goal: Value(x)
Answer: 12
Theorem sequence:
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, AB=18, BC=x, CA=y, ∠ABC=60°, triangle BCA is a right triangle. Find the value of x.
问题：如图所示，AB=18，BC=x，CA=y，∠ABC=60°，∠BCA是直角△BCA的直角。求x的值。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),18)
Equal(LengthOfLine(BC),x)
Equal(LengthOfLine(CA),y)
Equal(MeasureOfAngle(ABC),60)
RightTriangle(BCA)
Goal: Value(x)
Answer: 9
Theorem sequence:
sine_theorem(1,ABC)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, AB=30, CD=30, the measure of arc XCZ is 40, the center of ⊙X is X, AM⊥YM, DN is perpendicular to ZN. Find the measure of arc XCD.
问题：如图所示，AB=30，CD=30，⌒XCZ的角度为40，X是⊙X的圆心，AM⊥YM，DN⊥ZN。求弧XCD的角度。
Construction (CDL):
Shape(XBY,YM,MB)
Shape(XYA,AM,MY)
Shape(XAC,CN,NX,XM,MA)
Shape(XCZ,ZN,NC)
Shape(XZD,DN,NZ)
Shape(XDB,BM,MX,XN,ND)
Collinear(AMB)
Collinear(YMX)
Collinear(CND)
Collinear(XNZ)
Cocircular(X,ACZDBY)
Conditions (CDL):
Equal(LengthOfLine(AB),30)
Equal(LengthOfLine(CD),30)
Equal(MeasureOfArc(XCZ),40)
IsCentreOfCircle(X,X)
PerpendicularBetweenLine(AM,YM)
PerpendicularBetweenLine(DN,ZN)
Goal: Value(MeasureOfArc(XCD))
Answer: 80
Theorem sequence:
adjacent_complementary_angle(1,DNZ,ZNC)
adjacent_complementary_angle(1,ZNC,CNX)
circle_property_chord_perpendicular_bisect_arc(1,XCD,XNZ)
congruent_arc_judgment_length_equal(1,XCZ,XZD)
congruent_arc_property_measure_equal(1,XCZ,XZD)
arc_addition_measure(1,XCZ,XZD)
Problem: As shown in the diagram, AC=9, BA=9, ∠BAC=60°. Find the length of line BC.
问题：如图所示，AC=9，BA=9，∠BAC=60°。求直线BC的长度。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),9)
Equal(LengthOfLine(BA),9)
Equal(MeasureOfAngle(BAC),60)
Goal: Value(LengthOfLine(BC))
Answer: 9
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_property_angle_equal(1,ACB)
triangle_property_angle_sum(1,ACB)
sine_theorem(1,BAC)
Problem: As shown in the diagram, AB=5*x, CB=3*y-4, DA=29, DC=25, ∠ADB=34°, ∠CAD=49°, ∠DCA=54°, BADC is a parallelogram. Find the measure of ∠DFA.
问题：如图所示，AB=5*x，CB=3*y-4，DA=29，DC=25，∠ADB=34°，∠CAD=49°，∠DCA=54°，四边形BADC是平行四边形。求∠DFA的大小。
Construction (CDL):
Shape(BA,AF,FB)
Shape(FA,AD,DF)
Shape(FD,DC,CF)
Shape(FC,CB,BF)
Collinear(BFD)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(AB),5*x)
Equal(LengthOfLine(CB),3*y-4)
Equal(LengthOfLine(DA),29)
Equal(LengthOfLine(DC),25)
Equal(MeasureOfAngle(ADB),34)
Equal(MeasureOfAngle(CAD),49)
Equal(MeasureOfAngle(DCA),54)
Parallelogram(BADC)
Goal: Value(MeasureOfAngle(DFA))
Answer: 97
Theorem sequence:
triangle_property_angle_sum(1,FAD)
Problem: As shown in the diagram, ∠YXW=30°, WY⊥ZY, quadrilateral WYZX is a rectangle. Find the measure of ∠ZXA.
问题：如图所示，∠YXW=30°，WY⊥ZY，四边形WYZX是长方形。求∠ZXA的大小。
Construction (CDL):
Shape(WY,YA,AW)
Shape(AY,YZ,ZA)
Shape(AZ,ZX,XA)
Shape(AX,XW,WA)
Collinear(WAZ)
Collinear(YAX)
Conditions (CDL):
Equal(MeasureOfAngle(YXW),30)
PerpendicularBetweenLine(WY,ZY)
Rectangle(WYZX)
Goal: Value(MeasureOfAngle(ZXA))
Answer: 60
Theorem sequence:
angle_addition(1,ZXA,AXW)
Problem: As shown in the diagram, ∠GBJ=26°, ∠KBF=88°, the center of circle B is B. Find the measure of ∠JHG.
问题：如图所示，∠GBJ=26°，∠KBF=88°，B是⊙B的圆心。求∠JHG的大小。
Construction (CDL):
Shape(HG,BJG,JH)
Shape(BJG,GB,BJ)
Shape(BG,GF,FB)
Shape(JB,BK,KJ)
Shape(BGF,FG)
Shape(BKJ,JK)
Shape(KB,BF,BFK)
Collinear(KJH)
Collinear(FGH)
Cocircular(B,JGFK)
Conditions (CDL):
Equal(MeasureOfAngle(GBJ),26)
Equal(MeasureOfAngle(KBF),88)
IsCentreOfCircle(B,B)
Goal: Value(MeasureOfAngle(JHG))
Answer: 31
Theorem sequence:
arc_property_center_angle(1,BJG,B)
arc_property_center_angle(1,BFK,B)
circle_property_circular_power_segment_and_segment_angle(1,HJK,HGF,B)
Problem: As shown in the diagram, GF=HG, HJ=JF, ∠FJH=72°, ∠HGF=128°, GF and GH are one pair of adjacent sides of the kite GFJH. Find the measure of ∠GFJ.
问题：如图所示，GF=HG，HJ=JF，∠FJH=72°，∠HGF=128°，GF和GH是风筝形GFJH的一组临边。求∠GFJ的大小。
Construction (CDL):
Shape(FJ,JH,HG,GF)
Conditions (CDL):
Equal(LengthOfLine(GF),LengthOfLine(HG))
Equal(LengthOfLine(HJ),LengthOfLine(JF))
Equal(MeasureOfAngle(FJH),72)
Equal(MeasureOfAngle(HGF),128)
Kite(GFJH)
Goal: Value(MeasureOfAngle(GFJ))
Answer: 80
Theorem sequence:
kite_property_opposite_angle_equal(1,GFJH)
quadrilateral_property_angle_sum(1,FJHG)
Problem: As shown in the diagram, AC=18*sqrt(3), BA=x, CB=18*sqrt(3), ∠BAC=45°, AC and CB are the legs of the right △ ACB. Find the value of x.
问题：如图所示，AC=18*sqrt(3)，BA=x，CB=18*sqrt(3)，∠BAC=45°，AC和CB是直角△ACB的两个直角边。求x的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),18*sqrt(3))
Equal(LengthOfLine(BA),x)
Equal(LengthOfLine(CB),18*sqrt(3))
Equal(MeasureOfAngle(BAC),45)
RightTriangle(ACB)
Goal: Value(x)
Answer: 18*sqrt(6)
Theorem sequence:
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, AB=10, BD=5, BD⊥AD, CA⊥BA. Find the length of line BC.
问题：如图所示，AB=10，BD=5，BD⊥AD，CA⊥BA。求直线BC的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BD),5)
PerpendicularBetweenLine(BD,AD)
PerpendicularBetweenLine(CA,BA)
Goal: Value(LengthOfLine(BC))
Answer: 20
Theorem sequence:
line_addition(1,BD,DC)
adjacent_complementary_angle(1,BDA,ADC)
right_triangle_judgment_angle(1,BDA)
right_triangle_judgment_angle(1,ADC)
right_triangle_judgment_angle(1,CAB)
right_triangle_property_pythagorean(1,BDA)
right_triangle_property_pythagorean(1,ADC)
right_triangle_property_pythagorean(1,CAB)
Problem: As shown in the diagram, AC=11, BH=10, DB=8, the height of BCAD is BH, BH⊥AH, quadrilateral BCAD is a trapezoid. Find the area of quadrilateral BCAD.
问题：如图所示，AC=11，BH=10，DB=8，BCAD的高为BH，BH⊥AH，BCAD是梯形。求四边形BCAD的面积。
Construction (CDL):
Shape(BC,CH,HB)
Shape(BH,HA,AD,DB)
Collinear(CHA)
Conditions (CDL):
Equal(LengthOfLine(AC),11)
Equal(LengthOfLine(BH),10)
Equal(LengthOfLine(DB),8)
IsAltitudeOfQuadrilateral(BH,BCAD)
PerpendicularBetweenLine(BH,AH)
Trapezoid(BCAD)
Goal: Value(AreaOfQuadrilateral(BCAD))
Answer: 95
Theorem sequence:
trapezoid_area_formula(1,BCAD)
Problem: As shown in the diagram, AC is the diameter of ⊙D. Find the measure of ∠ABC.
问题：如图所示，AC是⊙D的直径。求∠ABC的大小。
Construction (CDL):
Shape(DAB,BA)
Shape(DBC,CB)
Shape(AB,BC,CD,DA)
Shape(AD,DC,DCA)
Collinear(ADC)
Cocircular(D,ABC)
Conditions (CDL):
IsDiameterOfCircle(AC,D)
Goal: Value(MeasureOfAngle(ABC))
Answer: 90
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,D)
Problem: As shown in the diagram, AC=8, AD=4, CB=10, HC=4, HD=8, AC⊥BC, DA⊥CA, DH⊥BH. Find the perimeter of quadrilateral BDAC.
问题：如图所示，AC=8，AD=4，CB=10，HC=4，HD=8，AC垂直于BC，DA垂直于CA，DH垂直于BH。求BDAC的周长。
Construction (CDL):
Shape(BD,DH,HB)
Shape(HD,DA,AC,CH)
Collinear(BHC)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(AD),4)
Equal(LengthOfLine(CB),10)
Equal(LengthOfLine(HC),4)
Equal(LengthOfLine(HD),8)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(DA,CA)
PerpendicularBetweenLine(DH,BH)
Goal: Value(PerimeterOfQuadrilateral(BDAC))
Answer: 32
Theorem sequence:
line_addition(1,BH,HC)
right_triangle_judgment_angle(1,DHB)
right_triangle_property_pythagorean(1,DHB)
quadrilateral_perimeter_formula(1,BDAC)
Problem: As shown in the diagram, QR=20, TS=12, ∠QRS=45°, ∠STQ=120°, A is the midpoint of segment TQ, B bisects segment SR, TQRS is a trapezoid. Find the measure of ∠RST.
问题：如图所示，QR=20，TS=12，∠QRS=45°，∠STQ=120°，A是线段TQ的中点，B平分线段SR，四边形TQRS是梯形。求∠RST的大小。
Construction (CDL):
Shape(AQ,QR,RB,BA)
Shape(TA,AB,BS,ST)
Collinear(TAQ)
Collinear(SBR)
Conditions (CDL):
Equal(LengthOfLine(QR),20)
Equal(LengthOfLine(TS),12)
Equal(MeasureOfAngle(QRS),45)
Equal(MeasureOfAngle(STQ),120)
IsMidpointOfLine(A,TQ)
IsMidpointOfLine(B,SR))
Trapezoid(TQRS)
Goal: Value(MeasureOfAngle(RST))
Answer: 135
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,RQ,ST)
Problem: As shown in the diagram, ∠ABD=45°, ∠BCD=70°, ∠DAB=40°. Find the measure of ∠BDA.
问题：如图所示，∠ABD=45°，∠BCD=70°，∠DAB=40°。求∠BDA的大小。
Construction (CDL):
Shape(BC,CD,DB)
Shape(BD,DA,AB)
Collinear(CDA)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),45)
Equal(MeasureOfAngle(BCD),70)
Equal(MeasureOfAngle(DAB),40)
Goal: Value(MeasureOfAngle(BDA))
Answer: 95
Theorem sequence:
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, the measure of ⌒CAB is 180. Find the measure of ∠BDA.
问题：如图所示，⌒CAB的角度为180。求∠BDA的大小。
Construction (CDL):
Shape(BD,DA)
Collinear(ACB)
Cocircular(C,ABD)
Conditions (CDL):
Equal(MeasureOfArc(CAB),180)
Goal: Value(MeasureOfAngle(BDA))
Answer: 90
Theorem sequence:
arc_property_circumference_angle_external(1,CAB,D)
Problem: As shown in the diagram, Add(MeasureOfAngle(DBA)=MeasureOfAngle(EBC)), ∠DBA=47°. Find the measure of ∠ABE.
问题：如图所示，Add(MeasureOfAngle(DBA)=MeasureOfAngle(EBC))，∠DBA=47°。求∠ABE的大小。
Construction (CDL):
Shape(DB,BA)
Shape(AB,BE)
Shape(EB,BC)
Collinear(CBD)
Conditions (CDL):
Equal(Add(MeasureOfAngle(DBA),MeasureOfAngle(EBC)),90)
Equal(MeasureOfAngle(DBA),47)
Goal: Value(MeasureOfAngle(ABE))
Answer: 90
Theorem sequence:
flat_angle(1,DBC)
angle_addition(1,DBA,ABC)
angle_addition(1,ABE,EBC)
Problem: As shown in the diagram, AE=5, BA=x, DC=10, EC=8, ∠ACD=∠CAB, ∠BEA=∠DEC, DC∥AB. Find the length of line AB.
问题：如图所示，AE=5，BA=x，DC=10，EC=8，∠ACD=∠CAB，∠BEA=∠DEC，DC平行于AB。求直线AB的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EC,CD,DE)
Collinear(AEC)
Collinear(BED)
Conditions (CDL):
Equal(LengthOfLine(AE),5)
Equal(LengthOfLine(BA),x)
Equal(LengthOfLine(DC),10)
Equal(LengthOfLine(EC),8)
Equal(MeasureOfAngle(ACD),MeasureOfAngle(CAB))
Equal(MeasureOfAngle(BEA),MeasureOfAngle(DEC))
ParallelBetweenLine(DC,AB)
Goal: Value(LengthOfLine(AB))
Answer: 25/4
Theorem sequence:
similar_triangle_judgment_aa(1,BEA,DEC)
similar_triangle_property_line_ratio(1,BEA,DEC)
similar_triangle_property_line_ratio(1,EAB,ECD)
Problem: As shown in the diagram, AD=10, CA=x, DE=8, EB=18, EBCA is a right trapezoid, DE and EA are the legs of the right △ DEA. Find the value of x.
问题：如图所示，AD=10，CA=x，DE=8，EB=18，EA和BC是直角梯形EBCA的平行边，三角形DEA为直角三角形。求x的值。
Construction (CDL):
Shape(DE,EA,AD)
Shape(EB,BC,CA,AE)
Collinear(DEB)
Collinear(DAC)
Conditions (CDL):
Equal(LengthOfLine(AD),10)
Equal(LengthOfLine(CA),x)
Equal(LengthOfLine(DE),8)
Equal(LengthOfLine(EB),18)
RightTrapezoid(EBCA)
RightTriangle(DEA)
Goal: Value(x)
Answer: 45/2
Theorem sequence:
similar_triangle_judgment_aa(1,ADE,CDB)
similar_triangle_property_line_ratio(1,ADE,CDB)
similar_triangle_property_line_ratio(1,EAD,BCD)
line_addition(1,DE,EB)
line_addition(1,DA,AC)
Problem: As shown in the diagram, AX=6*x, BA=z, BX=36, CB=y, XC=x, AX and XB are the legs of the right △ AXB, triangle CBA is a right triangle. Find the value of z.
问题：如图所示，AX=6*x，BA=z，BX=36，CB=y，XC=x，三角形AXB为直角三角形，CB和BA是直角三角形CBA的两个直角边。求z的值。
Construction (CDL):
Shape(BA,AX,XB)
Shape(BX,XC,CB)
Collinear(AXC)
Conditions (CDL):
Equal(LengthOfLine(AX),6*x)
Equal(LengthOfLine(BA),z)
Equal(LengthOfLine(BX),36)
Equal(LengthOfLine(CB),y)
Equal(LengthOfLine(XC),x)
RightTriangle(AXB)
RightTriangle(CBA)
Goal: Value(z)
Answer: 36*sqrt(7)
Theorem sequence:
line_addition(1,AX,XC)
adjacent_complementary_angle(1,AXB,BXC)
right_triangle_judgment_angle(1,BXC)
right_triangle_property_pythagorean(1,CBA)
right_triangle_property_pythagorean(1,AXB)
right_triangle_property_pythagorean(1,BXC)
Problem: As shown in the diagram, AE=12, AE=BE, DE=11, DE=CE, BDAC is a rhombus. Find the area of BDAC.
问题：如图所示，AE=12，AE=BE，DE=11，DE=CE，四边形BDAC是菱形。求四边形BDAC的面积。
Construction (CDL):
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Shape(BE,EC,CB)
Shape(EA,AC,CE)
Collinear(DEC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AE),12)
Equal(LengthOfLine(AE),LengthOfLine(BE))
Equal(LengthOfLine(DE),11)
Equal(LengthOfLine(DE),LengthOfLine(CE))
Rhombus(BDAC)
Goal: Value(AreaOfQuadrilateral(BDAC))
Answer: 264
Theorem sequence:
line_addition(1,DE,EC)
line_addition(1,AE,EB)
kite_area_formula_diagonal(1,BDAC)
Problem: As shown in the diagram, RT=3, SR=5, TS=4, RT⊥TT. Find the value of sin(TS).
问题：如图所示，RT=3，SR=5，TS=4，RT⊥TT。求sin(TS)的值。
Construction (CDL):
Shape(TS,SR,RT)
Conditions (CDL):
Equal(LengthOfLine(RT),3)
Equal(LengthOfLine(SR),5)
Equal(LengthOfLine(TS),4)
PerpendicularBetweenLine(RT,TS)
Goal: Value(Sin(MeasureOfAngle(TSR)))
Answer: 3/5
Theorem sequence:
cosine_theorem(1,SRT)
Problem: As shown in the diagram, JL=x+12, KJ=7*x, KJ=JL, LK=11*x-8, LK=KJ. Find the length of line JL.
问题：如图所示，JL=x+12，KJ=7*x，KJ=JL，LK=11*x-8，LK=KJ。求直线JL的长度。
Construction (CDL):
Shape(KJ,JL,LK)
Conditions (CDL):
Equal(LengthOfLine(JL),x+12)
Equal(LengthOfLine(KJ),7*x)
Equal(LengthOfLine(KJ),LengthOfLine(JL))
Equal(LengthOfLine(LK),11*x-8)
Equal(LengthOfLine(LK),LengthOfLine(KJ))
Goal: Value(LengthOfLine(JL))
Answer: 14
Theorem sequence:
Problem: As shown in the diagram, the measure of ⌒SRT is 93, the radius of ⊙S is 15. Find the area of the sector SRT.
问题：如图所示，弧SRT的角度为93，圆S的半径为15。求扇形SRT的面积。
Construction (CDL):
Shape(SR,SRT,TS)
Shape(ST,STR,RS)
Cocircular(S,RT)
Conditions (CDL):
Equal(MeasureOfArc(SRT),93)
Equal(RadiusOfCircle(S),15)
Goal: Value(AreaOfSector(SRT))
Answer: 465*pi/8
Theorem sequence:
sector_area_formula(1,SRT)
Problem: As shown in the diagram, ∠BCA=123-x°, ∠BCQ=42+x°, CP is the altitude of triangle CAB, CQ is the angle bisector of ∠BCA, R is the midpoint of segment AB. Find the measure of ∠QCA.
问题：如图所示，∠BCA=123-x°，∠BCQ=42+x°，CP是△CAB的高，CQ是∠BCA的角平分线，R平分线段AB。求∠QCA的大小。
Construction (CDL):
Shape(CA,AP,PC)
Shape(CP,PQ,QC)
Shape(CQ,QR,RC)
Shape(CR,RB,BC)
Collinear(APQRB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),123-x)
Equal(MeasureOfAngle(BCQ),42+x)
IsAltitudeOfTriangle(CP,CAB)
IsBisectorOfAngle(CQ,BCA)
IsMidpointOfLine(R,AB)
Goal: Value(MeasureOfAngle(QCA))
Answer: 55
Theorem sequence:
angle_addition(1,RCQ,QCA)
angle_addition(1,BCR,RCQ)
angle_addition(1,BCR,RCA)