\section*{Exercise 2 (4 points)}
The complex plane is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$. To every point $M$ with affixe $z$, we associate the point $M'$ with affixe
$$z' = -z^2 + 2z$$
The point $M'$ is called the image of point $M$.

\begin{enumerate}
  \item Solve in the set $\mathbb{C}$ of complex numbers the equation:
$$-z^2 + 2z - 2 = 0$$
Deduce the affixes of the points whose image is the point with affixe 2.

  \item Let $M$ be a point with affixe $z$ and $M'$ its image with affixe $z'$.

We denote $N$ the point with affixe $z_N = z^2$.\\
Show that $M$ is the midpoint of segment $[NM']$.

  \item In this question, we assume that the point $M$ with affixe $z$ belongs to the circle $\mathscr{C}$ with center O and radius 1. We denote $\theta$ an argument of $z$.\\
a. Determine the modulus of each of the complex numbers $z$ and $z_N$, as well as an argument of $z_N$ as a function of $\theta$.\\
b. On the figure given in the appendix on page 7, a point $M$ on the circle $\mathscr{C}$ has been represented.\\
Construct on this figure the points $N$ and $M'$ using a ruler and compass (leave the construction lines visible).\\
c. Let A be the point with affixe 1. What is the nature of triangle $AMM'$?
\end{enumerate}