\begin{propriete}[Identités remarquables]
%\begin{multicols}{3}
\begin{itemize}
  \item $\left\Vert{}\vv{u}+\vv{v}\right\Vert{}^2=\left(\vv{u}+\vv{v}\right)^2=\vv{u}^2+2\vv{u}\cdot \vv{v}+\vv{v}^2=\left\Vert{}\vv{u}\right\Vert{}^2+2\vv{u}\cdot \vv{v}+\left\Vert{}\vv{v}\right\Vert{}^2$
  \item $\left\Vert{}\vv{u}-\vv{v}\right\Vert{}^2=(\vv{u}-\vv{v})^2=\vv{u}^2-2\vv{u}\cdot \vv{v}+\vv{v}^2=\left\Vert{}\vv{u}\right\Vert{}^2-2\vv{u}\cdot \vv{v}+\left\Vert{}\vv{v}\right\Vert{}^2$
  \item $(\vv{u}+\vv{v})\cdot(\vv{u}-\vv{v})=\vv{u}^2-\vv{v}^2=\left\Vert{}\vv{u}\right\Vert{}^2-\left\Vert{}\vv{v}\right\Vert{}^2$
\end{itemize}
%\end{multicols}

\vspace{-\baselineskip}
\end{propriete}

\begin{preuve}Démontrons le premier point :
\vspace{-0.5\baselineskip}\begin{align*}
  \left\Vert{}\vv{u}+\vv{v}\right\Vert{}^2=(\vv{u}+\vv{v})^2&=(\vv{u}+\vv{v})\cdot(\vv{u}+\vv{v}) \\
   &= \vv{u}\cdot\vv{u}+\vv{u}\cdot\vv{v}+\vv{v}\cdot\vv{u}+\vv{v}\cdot\vv{v} \textrm{ par distributivité} \\
   &= \vv{u}^2+\vv{u}\cdot\vv{v}+\vv{u}\cdot\vv{v}+\vv{v}^2 \textrm{ par commutativité} \\
   &= \vv{u}^2+2\vv{u}\cdot \vv{v}+\vv{v}^2=\left\Vert{}\vv{u}\right\Vert{}^2+2\vv{u}\cdot \vv{v}+\left\Vert{}\vv{v}\right\Vert{}^2.
\end{align*}

%Les deux autres points se démontrent de manière analogue.
\end{preuve}
