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@mrrclb48z

平面図形「銀河最速！「令和7年度 京都府公立高等学校 中期選抜 第5問」様をsympyとFreeCADでやってみたい。

Last updated at Posted at 2025-04-14

・長文です。 中学数学の範囲外(三角関数)が含まれています。
・問題文は2次元ですが、３次元FreeCADのマクロで、XY平面上に作図しました。

オリジナル

Youtube

いつもの先生 様 (0:00〜7:20)　
銀河最速！「令和7年度 京都府公立高等学校 中期選抜 第5問
https://youtu.be/1w5b9XkEJGg

PDF

・保護されていない通信
26京都府 ＜R7年(2025年)全問PDF -4-
http://homepage1.canvas.ne.jp/ynaka/R7PDF/26kyoto.pdf#page=11
＜　数学苦手な中学生，応援します 様内
http://homepage1.canvas.ne.jp/ynaka/index.html

sympyのweb上での実行方法

SymPy Live Shellで。FreeCADのマクロは、以下で実行できません。
https://qiita.com/mrrclb48z/items/00dd08b0317069be9342#web%E4%B8%8A%E3%81%AE%E5%AE%9F%E8%A1%8C%E6%96%B9%E6%B3%95sympy-live-shell%E3%81%A7
　

最初に

・次を考えていませんでした。

sympyで(オリジナル 様の方法を参考に)

ver0.1

・>色々解き方がある。
　計算量は多いです。

# ver0.1 >色々解き方がある。
from sympy import *
var('BE,CD,EF,EG,EI,ID,DJ')
AB=3*sqrt(2)
BC=7
# degABC=45
# degBCD=60
# degBDC=90
BE =solve(Eq(1/sqrt(2),BE/AB),BE)[0]     #;print("#",BE)
CE =BC-BE                                 ;print("#",CE)
CD =solve(Eq(Rational(1,2),CD/BC),CD)[0] #;print("#",CD)
EF =solve(Eq(1/sqrt(3),EF/BE),EF)[0]     #;print("#",EF)
EG =solve(Eq(1/sqrt(3),EG/CE),EG)[0]     #;print("#",EG)
FG =EG-EF                                 ;print("#",FG)
CEG=Rational(1,2)*CE*EG                  #;print("#",CEG)
EI =solve(Eq(CE/CD     ,EI/ID),EI)[0]    #;print("#",EI) # 角の二等分線
DJ =solve(Eq(2 /sqrt(3),CD/DJ),DJ)[0]    #;print("#",DJ) 
CED=Rational(1,2)*CE*DJ                  #;print("#",CED)
CEI=CED*EI/(EI+ID)                       #;print("#",CEI)
FGH=sqrt(3)/4*FG**2                      #;print("#",FGH) # 正三角形
print("#",CEG-CEI-FGH)
# 4
# sqrt(3)/3
# 43*sqrt(3)/60

ver0.2

・正三角形の面積の計をFGの長さを計算せず に。面積比で、EIHFの面積計算。
　ありがたみがありませんでした。

# ver0.2 面積比で
from sympy import *
var('BCD,BD')
BD =Rational(7,2)*sqrt(3)
rep={BCD:Rational(1,2)*BD*BD/sqrt(3)}
BEF=BCD*((3            /BD)**2)             #;print("#",BEF)
ECG=BCD*((4            /BD)**2)              ;print("#",ECG ,",",ECG .subs(rep))
CDH=BCD*((Rational(7,2)/BD)**2)             #;print("#",CDH)
GFH=collect(BEF+ECG+CDH-BCD,BCD)            #;print("#",GFH)
ECI=BCD*Rational(4,7)*4/(4+Rational(7,2))    ;print("#",ECI ,",",ECI .subs(rep))
EIHF=ECG-GFH-ECI                             ;print("#",EIHF,",",EIHF.subs(rep))
# 64*BCD/147 , 8*sqrt(3)/3
# 32*BCD/105 , 28*sqrt(3)/15
# 86*BCD/735 , 43*sqrt(3)/60

ver0.3

・xy座標で。中途半端な感じがします。

# ver0.3 xy座標で。中途半端な感じがします。
from sympy import *
AB    =3*sqrt(2)
BC    =7
# degABC=45
# degBCD=60
# degBDC=90
# 
BE=   AB/sqrt(2)  #;print("#",BE)
CE=BC-BE          #;print("#",CE)
CD=BC/2           #;print("#",CD)
# 
B =  Point(0 ,0)
C =B+Point(BC,0)
D =C+Point(-CD/2     ,CD/2*sqrt(3))
A =B+Point(AB/sqrt(2),AB/sqrt(2)  )
# 
E =  Point(A.x ,0)
F =  Point(A.x ,BE    /sqrt(3))
G =  Point(A.x ,CE    /sqrt(3))
H =  Point(BC/2,(BC/2)/sqrt(3))                         # △BCHは二等辺三角形
I =E+(D-E)*C.distance(E)/(C.distance(E)+C.distance(D))  # 角の２等分線の定理  
FG=G.y-F.y
print("#",CE)
print("#",FG)
print("#",Polygon(E,I,H,F).area)
# 4
# sqrt(3)/3
# 43*sqrt(3)/60

sympyで(いつもの方法で)

・xy平面の三角関数で

ver0.4

# ver0.4 xy平面の三角関数で
from sympy import *
AB    ,BC           =3*sqrt(2),7
radABC,radBCD,radBDC=rad(45),rad(60),rad(90)
# 
B=  Point(0 ,0)
A=B+Point(AB*cos(radABC),AB*sin(radABC))
C=B+Point(BC,0)
D=C+Point(-BC/2*cos(radBCD),BC/2*sin(radBCD) )
E=  Point(A.x  ,0)
F=  Point(E.x  , E.x     *tan(pi-radBCD-radBDC))
G=  Point(E.x  ,(C.x-E.x)*tan(radBCD/2))
H=  Point(C.x/2,(C.x/2)  *tan(radBCD/2))               #△BCHは二等辺三角形
I=E+(D-E)*C.distance(E)/(C.distance(E)+C.distance(D))  # 角の２等分線の定理  
print("#",C.distance(E))
print("#",F.distance(G))
print("#",Polygon(E,I,H,F).area)
# 4
# sqrt(3)/3
# 43*sqrt(3)/60

ver0.5

・何が何でもTriangleで

# ver0.5 
from sympy import *
AB,BC=3*sqrt(2),7
trBEA=Triangle(asa=(45   ,AB/sqrt(2)   ,90))
trBCD=Triangle(asa=(90-60,BC           ,60))
trBEF=Triangle(asa=(30   ,AB/sqrt(2)   ,90))
trECG=Triangle(asa=(90   ,BC-AB/sqrt(2),30))
B,E,A=trBEA.vertices[0],trBEA.vertices[1],trBEA.vertices[2]
C,D  =                  trBCD.vertices[1],trBCD.vertices[2]
F    =                                    trBEF.vertices[2]
G    =E+                                  trECG.vertices[2]
H    =Line(C,G).intersection(Line(B,D))[0]   # ;print(H)     
I    =Line(C,G).intersection(Line(E,D))[0]   # ;print(H)     
print("#",C.distance(E))
print("#",F.distance(G))
print("#",Polygon (E,I,H,F).area)
print("#",Triangle(E,I,H)  .area+Triangle(E,H,F).area)
print("#",Triangle(E,C,G)  .area-Triangle(E,C,I).area-Triangle(F,H,G).area)
# 4
# sqrt(3)/3
# 43*sqrt(3)/60
# 43*sqrt(3)/60
# 43*sqrt(3)/60

ver0.6

・CAD風。
・＞AD//BCじゃないの罠すぎませんか？w　(Youtubeのコメント様より)

# ver0.6
from sympy import *
AB    =3*sqrt(2)
BC    =7
radABC=rad(45)
radBCD=rad(60)
radBDC=rad(90)
B=  Point(0 ,0)
C=B+Point(BC,0)                                   #;print(C)
A=  Point(AB,0)  .rotate( radABC,B)               #;print(A)
E=  Line (B ,C)  .projection(A)                   #;print(E)
D=  Point(BC/2,0).rotate(-radBCD,C)               #;print(D)
F=Line(E,A).intersection(Line(B,D))[0]            #;print(F)     
G=Line(E,A).intersection(
    Line(C,Point(BC/2,0).rotate(-radBCD/2,C)))[0] #;print(G)     
H=Line(C,G).intersection(Line(B,D))[0]            #;print(H)     
I=Line(C,G).intersection(Line(E,D))[0]            #;print(I)     
print("#",C.distance(E))
print("#",F.distance(G))
print("#",Polygon(E,I,H,F).area)
print("#",float(D.y-A.y))
# 4
# sqrt(3)/3
# 43*sqrt(3)/60
# 0.031088913245535265

FreeCADのマクロで作図

・問題文は2次元ですが、３次元FreeCADのマクロで、XY平面に作図しました。
・計算部分は、Ver.0.6 の コピー貼り付けです。

# 以下でます。　
# この機能は非推奨です。この機能を直接使用しないでください。
#   「make_linear_dimension」または「make_linear_dimension_obj」のいずれかを使用してください。
import FreeCAD
import Part
import DraftTools
import Draft
import Mesh
# # ##############################################
# ver0.6
from sympy import *
AB    =3*sqrt(2)
BC    =7
radABC=rad(45)
radBCD=rad(60)
radBDC=rad(90)
B=  Point(0 ,0)
C=B+Point(BC,0)                                   #;print(C)
A=  Point(AB,0)  .rotate( radABC,B)               #;print(A)
E=  Line (B ,C)  .projection(A)                   #;print(E)
D=  Point(BC/2,0).rotate(-radBCD,C)               #;print(D)
F=Line(E,A).intersection(Line(B,D))[0]            #;print(F)     
G=Line(E,A).intersection(
    Line(C,Point(BC/2,0).rotate(-radBCD/2,C)))[0] #;print(G)     
H=Line(C,G).intersection(Line(B,D))[0]            #;print(H)     
I=Line(C,G).intersection(Line(E,D))[0]            #;print(I)     
print("#",C.distance(E))
print("#",F.distance(G))
print("#",Polygon(E,I,H,F).area)
print("#",float(D.y-A.y))
# # 4
# # sqrt(3)/3
# # 43*sqrt(3)/60
# # 0.031088913245535265
############################################################################
# 3D作図 z=0 XY平面に作図しました。
# 2024-10-08 fontsize 追加しました。従来と互換性がありません。 
# 2024-10-20 HR 
# 2025-03-27 HL 
############################################################################
def myDimension(fontsize,myExtOvershoot,start_point, end_point,HVP):
    x1, y1 = start_point.args
    x2, y2 = end_point.args
    myFlipText = True
    if HVP=="PL":
       myP1  =Point(float(x1),float(y1))
       myP2  =Point(float(x2),float(y2))
       myFlipText = False
       myExtOvershoot1=abs(myExtOvershoot)
    elif HVP=="PR":
       myP1  =Point(float(x1),float(y1))
       myP2  =Point(float(x2),float(y2))
       myFlipText = False
       myExtOvershoot1=-abs(myExtOvershoot)
    elif HVP=="VL":
       if x1<x2 and y1<y2:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x1),float(y2))
          myExtOvershoot1= abs(myExtOvershoot)
          myFlipText = False
       elif x1<x2 and y1>y2:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x1),float(y2))
          myExtOvershoot1=-abs(myExtOvershoot)
       elif x1>x2 and y1<y2:
          myP1  =Point(float(x2),float(y1))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1= abs(myExtOvershoot)
          myFlipText = False
       else:
          myP1  =Point(float(x2),float(y1))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1=-abs(myExtOvershoot)
    elif HVP=="VR":
       if x1<x2 and y1<y2:
          myP1  =Point(float(x2),float(y1))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1=-abs(myExtOvershoot)
          myFlipText = False
       elif x1<x2 and y1>y2:
          myP1  =Point(float(x2),float(y1))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1= abs(myExtOvershoot)
       elif x1>x2 and y1<y2:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x1),float(y2))
          myExtOvershoot1=-abs(myExtOvershoot)
       else:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x1),float(y2))
          myExtOvershoot1= abs(myExtOvershoot)
    elif HVP=="HL":
       if x1 < x2 and y1<y2:
          myP1  =Point(float(x1),float(y2))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1= abs(myExtOvershoot)
          myFlipText = False
       elif x1 < x2 and y1>=y2:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x2),float(y1))
          myExtOvershoot1= abs(myExtOvershoot)
          myFlipText = False
          #myFlipText = True
       elif x1 > x2 and y1<y2:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x2),float(y1))
          myExtOvershoot1= abs(myExtOvershoot)
       else:
          myP1  =Point(float(x1),float(y2))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1= abs(myExtOvershoot)
    elif HVP=="HR":
       #if x1<x2 and y1<y2: # 2024-10-20 HR
       if x1<x2 and y1<=y2:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x2),float(y1))
          myExtOvershoot1=-abs(myExtOvershoot)
          myFlipText = False
       elif x1<x2 and y1>y2:
          myP1  =Point(float(x1),float(y2))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1=-abs(myExtOvershoot)
          myFlipText = False
       elif x1>x2 and y1<y2:
          myP1  =Point(float(x1),float(y2))
          myP2  =Point(float(x2),float(y2))
          myExtOvershoot1=-abs(myExtOvershoot)
       else:
          myP1  =Point(float(x1),float(y1))
          myP2  =Point(float(x2),float(y1))
          myExtOvershoot1=-abs(myExtOvershoot)
    else:  
      print("")
    myM   =myP1.midpoint(myP2)
    myT   =myM+myExtOvershoot1*((myP1-myM).unit).rotate( -pi/2 )
    dimension = Draft.makeDimension(
        FreeCAD.Vector(myP1.x, myP1.y, 0),           # 点1の座標
        FreeCAD.Vector(myP2.x, myP2.y, 0),           # 点2の座標
        FreeCAD.Vector(float(myT.x),float(myT.y),0)  # 矢印の位置を中央に設定
    )
    # Ext Overshootを設定
    dimension.ViewObject.ExtOvershoot = -float(myExtOvershoot)
    
    dimension.ViewObject.FlipText = myFlipText

    # 矢印のスタイルを設定
    dimension.ViewObject.ArrowType    = "Arrow"  # 矢印のタイプを "Arrow" に設定
    dimension.ViewObject.ArrowSize    = fontsize/5
    dimension.ViewObject.TextSpacing  = 0
    dimension.ViewObject.LineColor    = (0.0, 1.0, 0.0)  # 緑色
    dimension.ViewObject.FontSize     = fontsize  # フォントサイズを指定
def myXYZ2Txt_2D(A):
    return ""
def myXYZ2Txt_XY_2D(A):
    return '(' + str(A.x) + ',' + str(A.y) +  ')'
def myTxtXYZ_2D(fontsize,A,myWedgei):
    P5x=float(A.x)
    P5y=float(A.y)
    P5z=0.0
    p5 = FreeCAD.Vector(P5x, P5y, P5z)
    myText = Draft.makeText(myWedgei, p5)
    myText.Label = myWedgei
    FreeCADGui.ActiveDocument.ActiveObject.FontSize = str(fontsize,)+' mm'
    return
def myTxtXYZ_S_2D(fontsize,*xy_tx):
    for i in range(1,int(len(xy_tx)/2)+1):
        myTxtXYZ_2D(fontsize,xy_tx[2*i-2],xy_tx[2*i-1]+myXYZ2Txt_2D(xy_tx[2*i-2]) )
    return
def myTxtXYZ_XY_S_2D(fontsize,*xy_tx):
    for i in range(1,int(len(xy_tx)/2)+1):
        myTxtXYZ_2D(fontsize,xy_tx[2*i-2],xy_tx[2*i-1]+myXYZ2Txt_XY_2D(xy_tx[2*i-2]) )
    return

def myTxtXYZ_MoveY_2D(fontsize,MoveY,A,myWedgei):
    P5x=float(A.x)
    P5y=float(A.y)+MoveY
    P5z=0.0
    p5 = FreeCAD.Vector(P5x, P5y, P5z)
    myText = Draft.makeText(myWedgei, p5)
    myText.Label = myWedgei
    FreeCADGui.ActiveDocument.ActiveObject.FontSize = str(fontsize,)+' mm'
    return
def myTxtXYZ_XY_S_MoveY_2D(fontsize,MoveY,*xy_tx):
    for i in range(1,int(len(xy_tx)/2)+1):
        myTxtXYZ_MoveY_2D(fontsize,MoveY,xy_tx[2*i-2],xy_tx[2*i-1]+myXYZ2Txt_XY_2D(xy_tx[2*i-2]) )
    return
############################################################################
# 3D作図 z=0 XY平面に作図しました。
############################################################################
def myLine_2D(A,B):
    Ax,Ay,Az=float(A.x),float(A.y),0.0
    Bx,By,Bz=float(B.x),float(B.y),0.0
    pl = FreeCAD.Placement()
    pl.Rotation.Q = (0.4247081540122249, 0.17592004639554645, 0.33985110062924484, 0.8204732460821097)
    pl.Base = FreeCAD.Vector(-3.9166066876399563, -2.1670824762243774, 1.7495260956243028)
    points = [FreeCAD.Vector(Ax,Ay,Az), FreeCAD.Vector(Bx,By,Bz)]
    line = Draft.make_wire(points, placement=pl, closed=False, face=True, support=None)
    Draft.autogroup(line)
    return
def myLine_S_2D(*args):
    for i in range(1,len(args)):
        myLine_2D(args[i-1],args[i])
    return
def myLine_C_2D(*args):
    for i in range(1,len(args)):
        myLine_2D(args[i-1],args[i])
    myLine_2D(args[i],args[0])
    return
def myLine_H_2D(*args):
    for i in range(1,len(args)):
        myLine_2D(args[0],args[i])
    return
##################################################################################
myLine_C_2D(A,B,E,C,D)
myLine_S_2D(B,F,H,D)  
myLine_S_2D(A,G,F,E)  
myLine_S_2D(C,I,H,G)  
myLine_S_2D(E,I,D)  
myFontsize =0.3
myTxtXYZ_XY_S_2D(myFontsize,A,"A",B,"B",C,"C",D,"D",E,"E",F,"F",G,"G",H,"H",I,"I")
myOvershoot=myFontsize*4
myDimension(myFontsize,myOvershoot,B,A,"PL")
myDimension(myFontsize,myOvershoot,B,C,"HR")
myDimension(myFontsize,myOvershoot,A,D,"VR")
####################################################################################
doc = App.activeDocument()
#
# 2025-03-27 なんだか調子が悪い
# App.ActiveDocument.addObject("App::Origin", "Origin")
# App.ActiveDocumen!t.getObject('Origin').Visibility = True
origin = doc.addObject("App::Origin", "Origin")
# origin.Visibility = True
#
App.ActiveDocument.recompute()
Gui.activeDocument().activeView().viewAxonometric()
Gui.SendMsgToActiveView("ViewFit")

isometric方向。斜め方向から。
111.png
上空からです。平面図です。
・水平方向、鉛直方向、平行寸法は、override(手打ち)しています。
　無理数表示を勉強中です。
・角度寸法は、Measure Angularを使っています。
222.png

いつもの? sympyの実行環境と 参考のおすすめです。

(テンプレート)

いつもと違うおすすめです。

ver0.1

・The solvers module in SymPy implements methods for solving equations.
https://docs.sympy.org/latest/modules/solvers/solvers.html

ver0.2

collect() collects common powers of a term in an expression. For example
https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html#collect

ver0.3

The area of the polygon.
https://docs.sympy.org/latest/modules/geometry/polygons.html#sympy.geometry.polygon.Polygon.area

ver0.4

rotate(angle, pt=None)
https://docs.sympy.org/latest/modules/geometry/points.html#sympy.geometry.point.Point2D.rotate

ver0.5

class sympy.geometry.polygon.Triangle(*args, **kwargs)
https://docs.sympy.org/latest/modules/geometry/polygons.html#sympy.geometry.polygon.Triangle

ver0.6

The intersection with another geometrical entity.
https://docs.sympy.org/latest/modules/geometry/lines.html#sympy.geometry.line.LinearEntity.intersection

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