Le\; moulin\; de\; Von \;Aubel

A\left(  z_{1} \right) 

B\left(z_{2}  \right) 

D\left(z_{4}  \right) 

C\left(z_{3}  \right) 

M\left(z_{40}  \right) 

P\left(z_{41}  \right) 

N\left(  z_{43} \right) 

A''\left(  z_{15} \right) 

Q\left(  z_{42} \right) 

A'\left(  z_{9} \right) 

B''\left(z_{10}  \right) 

B'\left(z_{12}  \right) 

C''\left(z_{11}  \right) 

C'\left(z_{14}  \right) 

D''\left(z_{13}  \right) 

D'\left(z_{16}  \right) 

I

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Conjecture:\
En\;déplaçant \;l'un \;quelconque\;des\;sommets\;  du \;quadrilatère\ABCD,\;
 on \;vérifie \;de \;deux\; manières\; que\; les\;segments\;\left [MP  \right ] \
et\;\left [ QN \right ] \;restent\;isométriques \;et\; perpendiculaires

z_4 = (4.88, -0.46)
z_3 = (8.18, -1.86)
z_{13} = Rotate(z_3, -90*°, z_4)
P = Midpoint(z_3, z_{13})
z_2 = (6.18, 2.78)
z_{12} = Rotate(z_3, 90*°, z_2)
N = Midpoint(z_{12}, z_3)
z_1 = (3.36, 0.76)
z_{15} = Rotate(z_4, -90*°, z_1)
Q = Midpoint(z_{15}, z_4)
e = Line(Q, N)
z_{10} = Rotate(z_1, -90*°, z_2)
M = Midpoint(z_{10}, z_1)
c = Line(P, M)
poly1, z_5, z_6, z_7, z_8 = Polygon(z_1, z_2, z_3, z_4)
z_{9} = Rotate(z_2, 90*°, z_1)
z_{11} = Rotate(z_2, -90*°, z_3)
z_{14} = Rotate(z_4, 90*°, z_3)
z_{16} = Rotate(z_1, 90*°, z_4)
poly2, z_9, z_{17}, z_{18}, z_{19} = Polygon(z_1, z_2, z_{10}, z_{9})
poly3, z_{20}, z_{21}, z_{22}, z_{23} = Polygon(z_2, z_{12}, z_{11}, z_3)
poly4, z_{24}, z_{25}, z_{26}, z_{27} = Polygon(z_4, z_{13}, z_{14}, z_3)
poly5, z_{28}, z_{29}, z_{30}, z_{31} = Polygon(z_4, z_{16}, z_{15}, z_1)
a = Segment(Q, N)
b = Segment(M, P)
I = Intersect(c, e)
α = Angle(N, I, M)
texte3: "\left(QN   \right)∩\left(MP   \right)=\left\lbrace I \right\rbrace   \\
\;\;\;\;\;\;\;\;\;\widehat{MIN}=" + (LaTeX[α]) + ""
z_{41}: x(P) + (y(P) * ί)
z_{40}: x(M) + (y(M) * ί)
z_{42}: x(Q) + (y(Q) * ί)
n: z_{41} - z_{40}
z_{43}: x(N) + (y(N) * ί)
d: z_{43} - z_{42}
q: n / d
texte2: "\;\;\;\;\;\;\;\;\;\;\;1^{ère}\;méthode\\
\begin{array}{} Segment & \left [ QN \right ] &\left [ MP \right ]  \\ Longueur & " + (LaTeX[a]) + "&" + (LaTeX[b]) + " \\ \end{array} "
texte7: "2^{ème}\;méthode:avec \;les\; complxes\\
on\;a:\frac{z_P-z_M}{z_N-z_Q}=\frac{(" + (LaTeX[z_{41}]) + ")-(" + (LaTeX[z_{40}]) + ")}{(" + (LaTeX[z_{43}]) + ")-(" + (LaTeX[z_{42}]) + ")}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{" + (LaTeX[n]) + "}{" + (LaTeX[d]) + "}=" + (LaTeX[q]) + "=0 - i"
f = Segment(z_1, z_3)
g = Segment(z_4, z_2)
h = Segment(M, I)
i = Segment(Q, I)