O

A

\vec{u}

\vec{v}

M

P

N

L

K

Ex.2\;\; Métropole\;\; Juin \;\;2005

\mathscr{C}

Conjecture:méthode\;1\
D'après \;sa\; trace,\;le \;point\; N \;varie \;sur \;un\; cercle 

I

o = (0, 0)
a = (1, 0)
A = Midpoint(o, a)
ζ = Circle(A, o)
u = Vector(o, a)
j = (0, 1)
v = Vector(o, j)
B = Midpoint(o, j)
m = Point(ζ)
n = Rotate(a, 90*°, m)
p = Rotate(m, 270*°, a)
poly1, m_1, a_1, n_1, p_1 = Polygon(m, a, p, n)
l = Rotate(m, 90*°, o)
k = Rotate(o, 270*°, m)
poly2, o_1, m_2, k_1, l_1 = Polygon(o, m, k, l)
C = Point(ζ)
q_1: l / m
q_2: (p - (1 + ί)) / m
I = Midpoint(l, p)
q_3: (p + l) / 2
r: abs(q_3)
r': abs(p + l - 1) / 2
q_4: n - (p + l) / 2
s: abs(q_4)
o_2 = (0, 0)
a_2 = (1, 0)
A_1 = Midpoint(o_2, a_2)
ζ_1 = Circle(A_1, o_2)
u_1 = Vector(o_2, a_2)
m_3 = Point(ζ_1)
n_2 = Rotate(a_2, 90*°, m_3)
p_2 = Rotate(m_3, 270*°, a_2)
poly1_1, m_4, a_3, n_3, p_3 = Polygon(m_3, a_2, p_2, n_2)
l_2 = Rotate(m_3, 90*°, o_2)
C_1 = Point(ζ_1)
q_5: l_2 / m_3
q_6: (p_2 - (1 + ί)) / m_3
I_1 = Midpoint(l_2, p_2)
q_7: (p_2 + l_2) / 2
r_1: abs(q_7)
r'_1: abs(p_2 + l_2 - 1) / 2
q_8: n_2 - (p_2 + l_2) / 2
s_1: abs(q_8)
texte14: "*\;\frac{l}{m}=\frac{" + (LaTeX[l]) + "}{" + (LaTeX[m]) + "}=" + (LaTeX[q_1]) + "\;\;alors\;\;l=im\\
*\;\frac{p-\left(  1+i \right) }{m}=\frac{(" + (LaTeX[p]) + ")-\left( 1+i  \right) }{" + (LaTeX[m]) + "}=" + (LaTeX[q_2]) + "\\
\;\;\;\;alors\;\;p=-im+1+i\\
*\;\frac{p+l}{2}=" + (LaTeX[q_3]) + "\;\;donc\;le \;milieu\;I\;de\left [LP  \right ] \;est\;fixe\;et\\
\left\vert \frac{p+l}{2} \right\vert \;
=\left\vert \frac{\left(  " + (LaTeX[p]) + " \right)+\left(  " + (LaTeX[l]) + " \right)  }{2} \right\vert=" + (LaTeX[r]) + " =\frac{\sqrt{2}}{2}\\
\;\left\vert \frac{p+l}{2}-\frac{1}{2} \right\vert =\left\vert   \frac{p+l-1}{2} \right\vert=\left\vert  \frac{\left(  " + (LaTeX[p]) + " \right)+\left( " + (LaTeX[l]) + "  \right)-1  }{2}  \right\vert  =" + (LaTeX[r']) + "\\"
texte14_1: "Conjecture:méthode \;2\\
*\;\frac{l}{m}=\frac{" + (LaTeX[l_2]) + "}{" + (LaTeX[m_3]) + "}=" + (LaTeX[q_5]) + "\;\;alors\;\;l=im\\
*\;\frac{p-\left(  1+i \right) }{m}=\frac{(" + (LaTeX[p_2]) + ")-\left( 1+i  \right) }{" + (LaTeX[m_3]) + "}=" + (LaTeX[q_6]) + "\\
\;\;\;\;alors\;\;p=-im+1+i\\
*\;\frac{p+l}{2}=" + (LaTeX[q_7]) + "\;\;donc\;le \;milieu\;I\;de\left [LP  \right ] \;est\;fixe\\
*\;\left\vert n-\frac{p+l}{2}  \right\vert =\left\vert \left(  " + (LaTeX[n_2]) + " \right)-\frac{\left(  " + (LaTeX[p_2]) + " \right)+\left( " + (LaTeX[l_2]) + " \right)  }{2}  \right\vert  =" + (LaTeX[s_1]) + "=\frac{\sqrt{2}}{2}\\
ainsi \;la\; distance \;de\; I\; à\; N \;est \;fixe\; pour\;toute\; position \;de\; M \;sur\; \mathscr{C}\\
alors\; N\; varie\; sur\; le \;cercle\; de\; centre\; I \;et \;de \;rayon \frac{\sqrt{2}}{2}"