Let $ABC$ be a triangle such that $AB <AC$, $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$, then $B'$ is the point of tangency between $\omega_b$ and $(AC)$. Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$, then $C'$ is the point of tangency between $\omega_c$ and $(AB)$. Finally, let $I$ be the center of $\omega$ and $X$ the point of $\Gamma$ such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.
Problem
Source: The Francophone MO Juniors P1
Tags: congruent triangles, congruent, geometry, excircles, circumcircle, Francophone