入試問題 sympy FreeCAD Python

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。
私の解答の方法は、中学生の学習指導要領の範囲外です。 ベクトル成分表示です。
後半は、正弦定理を使っています。

オリジナル

数学を数楽に様　
youtube　(10:11) 大問３(３)

東進様
https://www.toshin.com/koukou_nyushi/files/2024/sugaku_q/chiba_sugaku_q.pdf#page=9
＜2024年全国公立高校入試解答速報｜大学受験の予備校・塾東進
https://www.toshin.com/koukou_nyushi/

sympyで(オリジナル様のやり方で)　できました。

・ver0.1
＞直角三角形の相似がたくさんある。
＞相似はかくれんぼ。
＞2つ45°
＞探せばいい

# ver0.1
from sympy import *
var('x,EBF',real=True,nonnegative=True)
x  =solve(Eq((15+x)/6,9/x),x)[0]             #;print("# x  =",x)
CF =sqrt (x**2+6**2)                         #;print("# CF =",CF)
EBF=solve(Eq(CF**2/15**2,6*x/2/EBF),EBF)[0]   ;print("# EBF=",EBF)
# EBF= 45

sympyで(いつものやり方で)

・ver0.2
・arctanでできませんでした。

# ver0.2
# from sympy import *
# var('x',real=True,nonnegative=True)
# y=(atan(9/x)+atan(6/x))*180/pi
# sol=solveset( Eq(y,45.0),x,Interval(17,19)   )    ;print("#",sol)
# # 
# # from sympy.plotting import plot
# # plot(0,45,y,(x,0,30))
# # ConditionSet(x, Eq(180*atan(6/x) + 180*atan(9/x) - 45.0*pi, 0), Interval(17, 19))

・ver0.3
・できました。正弦定理で。
・ベクトルの成分表示です。
・プログラム後半は、作図用です。三角形を回転しています。
Return the non-reflex angle formed by rays emanating from the origin with directions the same as the direction vectors of the linear entities.
https://docs.sympy.org/latest/modules/geometry/lines.html#sympy.geometry.line.LinearEntity.angle_between
Rotate angle radians counterclockwise about Point pt.
https://docs.sympy.org/latest/modules/geometry/points.html#sympy.geometry.point.Point2D.rotate

# ver0.3
from sympy import *
var('R',real=True,nonnegative=True)
deg,AD,DC=45,9,6
R =solve( Eq( (AD+DC)/sin(deg/180*pi),2*R),R)[0]
ax=Rational(1,2)*(AD+DC)
dx= ax-AD
O =Point(  0 ,0)
A =Point( ax ,-sqrt(R**2-ax**2))            
C =Point(-A.x,A.y)                          
D =Point( dx ,A.y)
B =Point( D.x,sqrt(R**2-(DC-ax)**2))         
E =Line(A,B).projection(C)
F =Line(B,D).intersection(Line(C,E))[0]
print("# EBF=",Triangle(E,B,F).area)
# EBF= 45
#
myRot=3/2*pi-Line(O,(C+B)/2).angle_between(Line(O,Point(1,0)))
A =A.rotate(myRot)                          
B =B.rotate(myRot)                          
C =C.rotate(myRot)                          
D =D.rotate(myRot)                          
E =E.rotate(myRot)                          
F =F.rotate(myRot)                          
# EBF= 45

・ver0.4
・極座標でできませんでした。

# ver0.4
# from sympy import *
# var('R,θA,θC',real=True,nonnegative=True)
# deg,AD,DC=45,9,6
# R=solve( Eq( (AD+DC)/sin(deg/180*pi),2*R),R)[0]
# A=Point( R*cos(θA), R*sin(θA))         
# B=Point(-R*cos(θC),-R*sin(θC))         
# C=Point( R*cos(θC),-R*sin(θC))         
# D=Line(A,C).projection(B)                                           #;print("# D  =",D)
# sol=solve( [Eq(A.distance(D),AD),Eq(D.distance(C),DC)] ,[θA,θC])   ;print("# sol=",sol)

FreeCADのマクロで作図

・計算部分は、コピー貼り付けです。

import FreeCAD
import Part
import DraftTools
import Draft
import Mesh
############################################################################
# 計算
# ver0.3
from sympy import *
var('R',real=True,nonnegative=True)
deg,AD,DC=45,9,6
R =solve( Eq( (AD+DC)/sin(deg/180*pi),2*R),R)[0]
ax=Rational(1,2)*(AD+DC)
dx= ax-AD
O =Point(  0 ,0)
A =Point( ax ,-sqrt(R**2-ax**2))            
C =Point(-A.x,A.y)                          
D =Point( dx ,A.y)
B =Point( D.x,sqrt(R**2-(DC-ax)**2))         
E =Line(A,B).projection(C)
F =Line(B,D).intersection(Line(C,E))[0]
print("# EBF=",Triangle(E,B,F).area)
#
myRot=3/2*pi-Line(O,(C+B)/2).angle_between(Line(O,Point(1,0)))
print("# myRot=",myRot*180/pi,float(myRot*180/pi) )
A =A.rotate(myRot)                          
B =B.rotate(myRot)                          
C =C.rotate(myRot)                          
D =D.rotate(myRot)                          
E =E.rotate(myRot)                          
F =F.rotate(myRot)                          
print("# B=",B )
print("# C=",C )
############################################################################
# 作図用
############################################################################
# 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=myCi.center.x
    y=myCi.center.y
    r=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 myXYZ2Txt_2D(A):
    return '(' + str(A.x) + ',' + str(A.y) +  ')'
    #return ""
def myTxtXYZ_2D(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 = '0.8 mm'
    FreeCADGui.ActiveDocument.ActiveObject.FontSize = '1.5 mm'
    return
def myTxtXYZ_S_2D(*xy_tx):
    for i in range(1,int(len(xy_tx)/2)+1):
        myTxtXYZ_2D(xy_tx[2*i-2],xy_tx[2*i-1]+myXYZ2Txt_2D(xy_tx[2*i-2]) )
    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
#######################################################################################
myLine_C_2D     (A,B,C)
myLine_S_2D     (B,D)
myLine_S_2D     (C,E)
myTxtXYZ_S_2D(O,"O",A,"A",B,"B",C,"C",D,"D",E,"E",F,"F")
myCircle_2D      (Circle(Point(0, 0), R)) 
#######################################################################################
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")