Exercise 2 (Candidates who have followed the specialization course)

We assume the following results are known:
\begin{itemize}
  \item the composition of two plane similarities is a plane similarity;
  \item the inverse transformation of a plane similarity is a plane similarity;
  \item a plane similarity that leaves three non-collinear points of the plane invariant is the identity of the plane.
\end{itemize}

\begin{enumerate}
  \item Let A, B and C be three non-collinear points in the plane and $s$ and $s'$ be two similarities of the plane such that $s(\mathrm{A}) = s'(\mathrm{A})$, $s(\mathrm{B}) = s'(\mathrm{B})$ and $s(\mathrm{C}) = s'(\mathrm{C})$. Show that $s = s'$.
  \item The complex plane is referred to the orthonormal frame $(\mathrm{O}, \vec{u}, \vec{v})$. We are given the points A with affix $2$, E with affix $1 + \mathrm{i}$, F with affix $2 + \mathrm{i}$ and G with affix $3 + \mathrm{i}$.\\
  a. Calculate the lengths of the sides of the triangles OAG and OEF. Deduce that these triangles are similar.\\
  b. Show that OEF is the image of OAG by an indirect similarity $S$, by determining the complex form of $S$.\\
  c. Let $h$ be the homothety with centre O and ratio $\frac{1}{\sqrt{2}}$. We set $\mathrm{A}' = h(\mathrm{A})$ and $\mathrm{G}' = h(\mathrm{G})$, and we call I the midpoint of $[\mathrm{EA}']$. We denote by $\sigma$ the orthogonal symmetry with axis (OI). Show that $S = \sigma \circ h$.
\end{enumerate}