(given to Yrvann Emzivat). Let $ABC$ be a triangle. A point $M$ on $[BC]$ is projected onto $B'$ and $C'$ on $(AC)$ and $(AB)$, respectively. For which point $M$ is the length $B'C'$ minimized?
. Given the right angles, the circle with diameter $[A M]$ passes through $B^{\prime}$ and $C^{\prime}$, and the angle $\widehat{B^{\prime} A C^{\prime}}$ is constant, equal to $\widehat{B A C}$.

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Thus $B^{\prime} C^{\prime}=A M \sin \widehat{B A C}$. It follows that $B^{\prime} C^{\prime}$ is minimal when $A M$ is minimal, which occurs when $M$ is the projection $H$ of $A$ onto $(B C)$: for any other position of $M$, the hypotenuse $A M$ of $A M H$ is strictly longer than $A H$. If $H$ does not lie on the segment $[B C]$, the minimum is achieved at one of the points $B$ or $C$, the one closest to $H$: this follows from the Pythagorean theorem.