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@mrrclb48z

2つの直角三角形「高校入試頻出問題 EB=?」をsympyとFreeCADでやってみた。

Last updated at Posted at 2024-11-10

・5通り理解できました。
・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

・YUUU0123 様 (0:00〜3:13)

sympyで(オリジナル 様の方法を参考に)

ver0.1 二等辺三角形です。

# ver0.1 二等辺三角形です。 
from sympy import *
BC,CD=1,3
AD=BC+CD
AC=sqrt(AD**2-CD**2) ;print("#",AC)
AB=sqrt(AC**2+BC**2) ;print("#",AB)
EB=AB*Rational(1,2)  ;print("#",EB)
# sqrt(7)
# 2*sqrt(2)
# sqrt(2)

ver0.2 相似です。Youtubeのコメント様より。

# ver0.2 相似です。 
from sympy import *
var('x',positive=True)
print("#",solve(Eq(2*x/1,(1+3)/x),x)[0])
# sqrt(2)

ver0.3 方べきの定理です。Youtubeのコメント様より。

# ver0.3 方べきの定理です。
from sympy import *
var('BE',positive=True)
BC,CD=1,3
BD=BC+CD
BA=2*BE
print("#",solve(Eq(BC*BD,BE*BA),BE)[0])
# sqrt(2)

sympyで(いつもの方法で)　

ver0.4 電卓です。

・√(2^2-1.5^2+0.5^2)=√2
・二等辺三角形を使わず、相似を使わず、変数を使わずです。

#  ver0.4 電卓です。
from sympy import *
print("#",sqrt(2**2-Rational(3/2)**2+Rational(1,2)**2))
# sqrt(2)

ver0.5 CADです。

・点Eは、円と直線CA左オフセット(x=1/2)との交点計算です。作図用です。

# ver0.5 CADです。
from sympy import *
var('x',positive=True)
BC,CD=1,3
B,C,D=map(Point,[(0,0),(BC,0),(BC+CD,0)])
x1wa2=Line(B,C).perpendicular_line(Segment(B,C).midpoint)
E    =Circle(Segment(B,D).midpoint,(BC+CD)*Rational(1,2)).intersection(x1wa2)[1]
print("#",B.distance(E))
# sqrt(2)

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.0.5 の コピー貼り付けです。

import FreeCAD
import Part
import DraftTools
import Draft
import Mesh
############################################################################
# ver0.5 CADです。
from sympy import *
var('x',positive=True)
BC,CD=1,3
B,C,D=map(Point,[(0,0),(BC,0),(BC+CD,0)])
x1wa2=Line(B,C).perpendicular_line(Segment(B,C).midpoint)
E    =Circle(Segment(B,D).midpoint,(BC+CD)*Rational(1,2)).intersection(x1wa2)[1]
print("#",B.distance(E))
# sqrt(2)
############################################################################
A=B+2*E
############################################################################
# 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 バグ対応済
############################################################################
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
       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:
          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。
############################################################################
# 円の作図　FrecCADのdocより
# https://wiki.freecad.org/Macro_Circle
def Freecad3D_circle(x=0.0,y=0.0,z=0.0,radius=0.0,diameter=0.0,circumference=0.0,area=0.0,startangle=0.0,endangle=0.0,arc=0.0,anglecenter=0.0,cord=0.0,arrow=0.0,center=0,placemObject=""):
    from math import sqrt, pi
    if placemObject == "":
        pl = FreeCAD.Placement()
        pl.Rotation = FreeCADGui.ActiveDocument.ActiveView.getCameraOrientation()   
        pl.Base = FreeCAD.Vector(x,y,z)
    else:
        pl = FreeCAD.Placement()
        pl = placemObject                                  # placement imposted
    if diameter != 0:                                      # with diameter
        radius = diameter / 2.0
    elif circumference != 0:                               # with circumference
        radius = (circumference / pi) / 2.0
    elif area != 0:                                        # with area
        radius =  sqrt((area / pi))
    elif (cord != 0) and (arrow != 0):                     # with cord and arrow
        radius = ((arrow**2) + (cord**2) / 4.0) / (arrow*2) 
    elif (arc != 0) and (anglecenter != 0):                # with arc and anglecenter central in degrees
        radius = ((360/anglecenter)*arc) / pi/2.0
        if endangle != 0:
            startangle  = endangle - anglecenter
        endangle   = anglecenter + startangle
        startangle  = endangle - anglecenter
    if radius != 0:
        try:
            Draft.makeCircle(radius,placement=pl,face=False,startangle=startangle,endangle=endangle,support=None)
            if center != 0:
                x=pl.Base.x
                y=pl.Base.y
                z=pl.Base.z
                Draft.makePoint(x,y,z)
        except Exception:
            App.Console.PrintError("Unexpected keyword argument" + "\n")
        App.ActiveDocument.recompute()
    else:
        App.Console.PrintMessage("Unexpected keyword argument" + "\n")
        App.Console.PrintMessage("circle(x,y,z,radius,diameter,circumference,area,startangle,endangle,[arc,anglecenter],[cord,arrow],center,placemObject)" + "\n")
        App.Console.PrintMessage("circle(radius=10.0,placemObject=App.Placement(App.Vector(11,20,30), App.Rotation(30,40,0), App.Vector(0,0,0)))" + "\n")
    return
def myCircle_2D(myCi):
    x=float(myCi.center.x)
    y=float(myCi.center.y)
    r=float(myCi.radius)
    Freecad3D_circle(
           x=float(x),y=float(y),z=0.0,
           radius=float(abs(r)),
           center=1,
           placemObject=App.Placement(App.Vector(float(x),float(y),0),
           App.Rotation(0,0,0),App.Vector(0,0,0)))
    return
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
##################################################################################
#myCircle_2D     (Circle(O,r_sol))
#myLine_C_2D     (A,B,C,D)
myLine_S_2D     (C,A,E,B,D,E)
#
#myFontsize =0.3
#myOvershoot=myFontsize*8
myFontsize =0.2
myOvershoot=myFontsize*4
#myTxtXYZ_XY_S_2D(myFontsize,A,"A",B,"B",C,"C",D,"D",E,"E")
myTxtXYZ_S_2D(myFontsize,A,"A",B,"B",C,"C",D,"D",E,"E")
myDimension(myFontsize,myOvershoot,B,E,"PL")
myDimension(myFontsize,myOvershoot,E,A,"PL")
myDimension(myFontsize,myOvershoot,B,C,"HR")
myDimension(myFontsize,myOvershoot,C,D,"HR")
####################################################################################
doc = App.activeDocument()
#App.ActiveDocument.addObject("App::Origin", "Origin")
#App.ActiveDocument.getObject('Origin').Visibility = True
App.ActiveDocument.recompute()
Gui.activeDocument().activeView().viewAxonometric()
Gui.SendMsgToActiveView("ViewFit")

isometric方向?です。(省略)
拡大図
souji.png

いつものリンク? sympyの実行環境と 参考のおすすめです。

(テンプレート)

・以下ができたら、助かります。指定と全部です

いつもと違うリンクのおすすめです。

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