The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D distinct with complex numbers $z _ { \mathrm { A } } , z _ { \mathrm { B } } , z _ { \mathrm { C } }$ and $z _ { \mathrm { D } }$ such that: $$\left\{ \begin{array} { l }
z _ { \mathrm { A } } + z _ { \mathrm { C } } = z _ { \mathrm { B } } + z _ { \mathrm { D } } \\
z _ { \mathrm { A } } + \mathrm { i } z _ { \mathrm { B } } = z _ { \mathrm { C } } + \mathrm { i } z _ { \mathrm { D } }
\end{array} \right.$$ Prove that the quadrilateral ABCD is a square.
\section*{Exercise 4}
\section*{Common to all candidates}
The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).\\
We consider the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D distinct with complex numbers $z _ { \mathrm { A } } , z _ { \mathrm { B } } , z _ { \mathrm { C } }$ and $z _ { \mathrm { D } }$ such that:
$$\left\{ \begin{array} { l }
z _ { \mathrm { A } } + z _ { \mathrm { C } } = z _ { \mathrm { B } } + z _ { \mathrm { D } } \\
z _ { \mathrm { A } } + \mathrm { i } z _ { \mathrm { B } } = z _ { \mathrm { C } } + \mathrm { i } z _ { \mathrm { D } }
\end{array} \right.$$
Prove that the quadrilateral ABCD is a square.