OFM 2020 J1

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.