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「生成AI開発の珍プレー好プレー大賞！（珍プレー多め）」
@mrrclb48z

メネラウスの定理。大人用「2024 青森公立大学前期【４】」(その3/4)をAI先生へ。sympyのいつもの方法で。

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Last updated at Posted at 2025-11-06

(その1/4)
(その2/4)
(その3/4) 本ページ
(その4/4)

・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。

(再出)オリジナル

(再出)web

数学入試問題 様へ。いつもお世話になっております。
https://kamelink.com/blog/2024/04/18/4-6-24-aomorikoritsudai-4/

sympyで(webオリジナル 様の方法を参考に)

ver2.0

・自力で?力技?
・＞△ABC でチェバの定理を用いると...　
・＞△AFC と直線 PE でメネラウスの定理を用いると...

# ver2.0
from sympy import *
var('BF_FC,AP_PF')
var('S,SA,SB,SC')
AB,CA=8,5
AD,CE=3,2
EA   =CA-CE
DB   =AB-AD
BF_FC=solve(Eq(BF_FC*CE/EA*AD/DB                                   ,1),BF_FC)[0]              #;print(BF_FC)
AP_PF=solve(Eq(AP_PF*numer(BF_FC)/(numer(BF_FC)+denom(BF_FC))*CE/EA,1),AP_PF)[0]               ;print("#",AP_PF)
DEF  =(S-(SA+SB+SC)).subs({SA:Rational(AD,AB)*Rational(AD,CA)                         *S, \
                           SB:Rational(CA,AB)*numer(BF_FC)/(numer(BF_FC)+denom(BF_FC))*S, \
                           SC:Rational(denom(BF_FC),denom(BF_FC)+numer(BF_FC))*Rational(CE,CA)                         *S  \
                          })                                                                
print("#",DEF.subs({S:Rational(1,2)*AB*CA*sin(60*pi/180)}))
# 21/10
# 15*sqrt(3)/7

sympyで(いつもの方法で)

ver3.0

・自力で?力技?
・trABCのBCを水平になるように回転して、trBCAを再定義しています。

# ver3.0
from sympy import *
AB,CA=8,5
AD,CE=3,2
trABC=Triangle(sas=(CA,60,AB))                   #;print(trABC)
BC   =trABC.args[1].distance(trABC.args[2])      #;print(BC)
trBCA=Triangle(sss=(BC,CA,AB))                   #;print(trBCA)
A,B,C=trBCA.args[2],trBCA.args[0],trBCA.args[1]
D    =A+AD*(B-A).unit
E    =C+CE*(A-C).unit
P    =Line(B,E).intersection(Line(C,D))[0]       #;print(P)
F    =Line(A,P).intersection(Line(B,C))[0]       #;print(F)
print("#",A.distance(P)/P.distance(F))
print("#",abs(Triangle(D,E,F).area))
# 21/10
# 15*sqrt(3)/7

(いつもの)FreeCADのマクロで作図

・自力で?力技?
・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.3.0 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
############################################################################
# ver3.0
from sympy import *
AB,CA=8,5
AD,CE=3,2
trABC=Triangle(sas=(CA,60,AB))                   #;print(trABC)
BC   =trABC.args[1].distance(trABC.args[2])      #;print(BC)
trBCA=Triangle(sss=(BC,CA,AB))                   #;print(trBCA)
A,B,C=trBCA.args[2],trBCA.args[0],trBCA.args[1]
D    =A+AD*(B-A).unit
E    =C+CE*(A-C).unit
P    =Line(B,E).intersection(Line(C,D))[0]       #;print(P)
F    =Line(A,P).intersection(Line(B,C))[0]       #;print(F)
print("#",A.distance(P)/P.distance(F))
print("#",abs(Triangle(D,E,F).area))
# 21/10
# 15*sqrt(3)/7
############################################################################
#B,C,A=BCA.args[0],BCA.args[1],BCA.args[2]
#E,D   =                   BED.args[1],BED.args[2]  
#F      =BCA.intersection(BED) [2]       ;print(F)
# F=Point(0,0)
############################################################################
# 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
# 2025-10-09 ChatGPT先生より\\\\\
#    dimension.ViewObject.ShowUnit     = False   # プロパティで単位を表示しないように設定
#    dimension.ViewObject.Decimals     = 0 
############################################################################
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
 
    # 2025-10-09 ChatGPT先生より
    dimension.ViewObject.ShowUnit     = False   # プロパティで単位を表示しないように設定
    dimension.ViewObject.Decimals     = 0
    return
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,D,B,F,C,E)
myLine_S_2D(A,P,F)
myLine_S_2D(C,P,D)
myLine_S_2D(B,P,E)
myFontsize =0.3
myTxtXYZ_XY_S_2D(myFontsize,A,"A",B,"B",C,"C",D,"D",E,"E",F,"F",P,"P")
myOvershoot=myFontsize*9
myDimension(myFontsize,myOvershoot                  ,B,A,"PL")
myDimension(myFontsize,myOvershoot-myFontsize,D,A,"PL")
myDimension(myFontsize,myOvershoot                  ,A,C,"PL")
myDimension(myFontsize,myOvershoot-myFontsize,E,C,"PL")
####################################################################################
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")

・(青色の)角度寸法は、CADの操作で追加。Part＞計測＞角度計算 です。
　角度寸法の表示は、勉強中です。
000.png

(いつもと違って本日の)ChatGPT先生へ。FreeCADのマクロのコンバートを教えて下さい。

質問

以下を作図するFreeCADのマクロを教えて下さい。
(ver3.0ソースコードを省略)

回答

# -*- coding: utf-8 -*-
# ver3.0 FreeCADマクロ
# SymPy版の構成:
# AB=8, CA=5, ∠A=60°, AD=3, CE=2
# P = BE ∩ CD, F = AP ∩ BC

import FreeCAD as App
import Draft, math

doc = App.newDocument("Triangle_Construction")

# --- 与条件 ---
AB, CA = 8, 5
AD, CE = 3, 2
angA = math.radians(60)

# --- 点A, B, C ---
A = App.Vector(0, 0, 0)
B = App.Vector(AB, 0, 0)
# Aから角60°でCA=5の位置にC
C = App.Vector(CA * math.cos(angA), CA * math.sin(angA), 0)

# --- 点D, E ---
D = A + (B - A).normalize().multiply(AD)
E = C + (A - C).normalize().multiply(CE)

# --- 点P = BE ∩ CD ---
def intersection(p1, p2, p3, p4):
    """2直線 p1p2, p3p4 の交点を求める"""
    x1, y1, x2, y2 = p1.x, p1.y, p2.x, p2.y
    x3, y3, x4, y4 = p3.x, p3.y, p4.x, p4.y
    den = (x1 - x2)*(y3 - y4) - (y1 - y2)*(x3 - x4)
    if abs(den) < 1e-10:
        return None
    px = ((x1*y2 - y1*x2)*(x3 - x4) - (x1 - x2)*(x3*y4 - y3*x4)) / den
    py = ((x1*y2 - y1*x2)*(y3 - y4) - (y1 - y2)*(x3*y4 - y3*x4)) / den
    return App.Vector(px, py, 0)

P = intersection(B, E, C, D)

# --- 点F = AP ∩ BC ---
F = intersection(A, P, B, C)

# --- 作図 ---
# 三角形ABC
Draft.makeLine(A, B)
Draft.makeLine(B, C)
Draft.makeLine(C, A)

# 補助線
Draft.makeLine(A, P)
Draft.makeLine(B, E)
Draft.makeLine(C, D)

# 点
for p, name in zip([A, B, C, D, E, P, F], ["A", "B", "C", "D", "E", "P", "F"]):
    Draft.makePoint(p)
    Draft.makeText([name], point=p + App.Vector(0.2, 0.2, 0))

App.ActiveDocument.recompute()

# --- 計算出力 ---
AP = (A - P).Length
PF = (P - F).Length
ratio = AP / PF

# 三角形DEFの面積
def tri_area(p1, p2, p3):
    return abs(0.5 * ((p2 - p1).x*(p3 - p1).y - (p2 - p1).y*(p3 - p1).x))

area_DEF = tri_area(D, E, F)

print("# AP/PF =", ratio)
print("# area(DEF) =", area_DEF)

# 結果（理論値）:
# AP/PF ≈ 21/10 = 2.1
# 面積 ≈ 15*sqrt(3)/7 ≈ 3.708

・ここであきらめました。続けて、質問すべきでした。
000.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。

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