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・以下を、BCを水平で解いて下さい。作図をお願いします。
・mathjax をChatGPT先生へ、丸投げです。
・(2025-11-06追加)FreeCADのマクロを教えて下さい。
・(2025-11-06追加)その3/3

(本日の)ChatGPT先生へ

質問

以下を、BCを水平で解いて下さい。作図をお願いします。
(mathjaxを省略)

回答

import math
import matplotlib.pyplot as plt
from math import sqrt

# 点の定義
B = (0.0, 0.0)
C = (7.0, 0.0)
A = (44.0/7.0, 20.0*math.sqrt(3)/7.0)
D = (55.0/14.0, 25.0*math.sqrt(3)/14.0)
E = (47.0/7.0, 8.0*math.sqrt(3)/7.0)

# 交点 P (計算済の値を使う)
P = (8225.0/1519.0, (1400.0*math.sqrt(3))/1519.0)
F = (5.0, 0.0)

# 描画
fig, ax = plt.subplots(figsize=(6,6))

# 三角形 ABC
ax.plot([A[0],B[0],C[0],A[0]],[A[1],B[1],C[1],A[1]],'-')
# 線分 BE, CD, AP
ax.plot([B[0],E[0]],[B[1],E[1]],'--')
ax.plot([C[0],D[0]],[C[1],D[1]],'--')
ax.plot([A[0],P[0],F[0]],[A[1],P[1],F[1]],':')

# 点のプロットとラベル
points = {'A':A,'B':B,'C':C,'D':D,'E':E,'P':P,'F':F}
for name,(x,y) in points.items():
    ax.scatter(x,y)
    ax.text(x+0.08,y+0.08,name)

ax.set_aspect('equal', adjustable='box')
ax.axhline(0,color='gray',linewidth=0.5)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_title('Triangle ABC with D on AB (AD=3), E on CA (CE=2), P=BE∩CD, F=AP∩BC')
plt.grid(True)
plt.savefig('triangle_DEF.png', dpi=200)
plt.show()

最終解答（要約）をコピーできるテキスト形式で。

最終解答（要約）

問題1:
AP : PF = 21 : 10

問題2:
[△DEF] = 15√3 / 7  （およそ 3.712）

(本日の)ChatGPT先生へ。FreeCADのマクロを教えて下さい。

質問

以下を、BCを水平で解いて下さい。FreeCADのマクロを教えて下さい。
(mathjaxを省略)

回答

# FreeCAD macro: Draw triangle ABC with BC horizontal, construct points D (on AB, AD=3) and E (on CA, CE=2),
# find intersection P of BE and CD, extend AP to meet BC at F, compute AP:PF and area of triangle DEF.
# How to run: Paste into FreeCAD Macro editor and run, or save as .FCMacro and run.

import FreeCAD, FreeCADGui
import Draft
import Part
import math

DOC_NAME = 'Triangle_DEF_Doc'
if FreeCAD.ActiveDocument is None or FreeCAD.ActiveDocument.Name != DOC_NAME:
    try:
        FreeCAD.closeDocument(DOC_NAME)
    except Exception:
        pass
    doc = FreeCAD.newDocument(DOC_NAME)
else:
    doc = FreeCAD.ActiveDocument

# Coordinates (BC horizontal):
B = FreeCAD.Vector(0.0, 0.0, 0.0)
C = FreeCAD.Vector(7.0, 0.0, 0.0)   # BC = 7 from law of cosines
Ax = 44.0/7.0
Ay = 20.0 * math.sqrt(3.0) / 7.0
A = FreeCAD.Vector(Ax, Ay, 0.0)

# Helper: linear interpolation
def lerp(P, Q, t):
    return FreeCAD.Vector(P.x + t*(Q.x - P.x), P.y + t*(Q.y - P.y), 0.0)

# D on AB with AD = 3 (AB = 8) -> fraction from A toward B is 3/8
D = lerp(A, B, 3.0/8.0)
# E on CA with CE = 2 (CA = 5) -> fraction from C toward A is 2/5
E = lerp(C, A, 2.0/5.0)

# 2D line intersection (p1->p2) with (q1->q2)
def line_intersection(p1, p2, q1, q2):
    # Solve p1 + s*(p2-p1) = q1 + t*(q2-q1)
    dx1 = p2.x - p1.x
    dy1 = p2.y - p1.y
    dx2 = q2.x - q1.x
    dy2 = q2.y - q1.y
    denom = dx1*dy2 - dy1*dx2
    if abs(denom) < 1e-12:
        return None
    s = ((q1.x - p1.x)*dy2 - (q1.y - p1.y)*dx2) / denom
    return FreeCAD.Vector(p1.x + s*dx1, p1.y + s*dy1, 0.0)

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

# Intersection F of line AP with BC (y=0) -> param t solving Ay + t*(Py-Ay) = 0
if P is None:
    FreeCAD.Console.PrintError('Lines BE and CD are parallel (no intersection)\n')
else:
    t = -A.y / (P.y - A.y)
    F = FreeCAD.Vector(A.x + t*(P.x - A.x), 0.0, 0.0)

    # create geometry: triangle edges and lines
    objs = []
    objs.append(Draft.makeLine(A, B))
    objs.append(Draft.makeLine(B, C))
    objs.append(Draft.makeLine(C, A))
    objs.append(Draft.makeLine(B, E))
    objs.append(Draft.makeLine(C, D))
    objs.append(Draft.makeLine(A, P))

    # markers for points
    for name, pt in [('A', A), ('B', B), ('C', C), ('D', D), ('E', E), ('P', P), ('F', F)]:
        s = Part.makeSphere(0.05, pt)
        part = doc.addObject('Part::Feature', 'pt_' + name)
        part.Shape = s
        txt = Draft.makeText([name], point=FreeCAD.Vector(pt.x + 0.08, pt.y + 0.08, 0))
        objs.append(part)
        objs.append(txt)

    doc.recompute()

    # compute distances
    def dist(P, Q):
        return math.hypot(P.x - Q.x, P.y - Q.y)

    AP = dist(A, P)
    PF = dist(P, F)
    ratio = AP / PF if PF != 0 else None

    # area of triangle DEF using shoelace
    def triangle_area(p, q, r):
        return abs((p.x*(q.y-r.y) + q.x*(r.y-p.y) + r.x*(p.y-q.y)) / 2.0)

    area_DEF = triangle_area(D, E, F)

    FreeCAD.Console.PrintMessage('Coordinates:\n')
    FreeCAD.Console.PrintMessage(f'A = ({A.x:.6f}, {A.y:.6f})\n')
    FreeCAD.Console.PrintMessage(f'B = ({B.x:.6f}, {B.y:.6f})\n')
    FreeCAD.Console.PrintMessage(f'C = ({C.x:.6f}, {C.y:.6f})\n')
    FreeCAD.Console.PrintMessage(f'D = ({D.x:.6f}, {D.y:.6f})\n')
    FreeCAD.Console.PrintMessage(f'E = ({E.x:.6f}, {E.y:.6f})\n')
    FreeCAD.Console.PrintMessage(f'P = ({P.x:.6f}, {P.y:.6f})\n')
    FreeCAD.Console.PrintMessage(f'F = ({F.x:.6f}, {F.y:.6f})\n')
    FreeCAD.Console.PrintMessage('\nResults:\n')
    if ratio is not None:
        FreeCAD.Console.PrintMessage(f'AP:PF = {ratio:.6f} -> exact 21:10\n')
    FreeCAD.Console.PrintMessage(f'Area(DEF) = {area_DEF:.12f} -> exact 15*sqrt(3)/7 (~{(15*math.sqrt(3)/7):.12f})\n')

    # fit view
    try:
        FreeCADGui.SendMsgToActiveView('ViewFit')
    except Exception:
        pass

・ここであきらめました。

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いつもの? sympyの実行環境と参考のおすすめです。

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