CAPS 2021 Problem 2

In an acute triangle $ABC$ , the incircle $\omega$ touches $BC$ at $D$ . Let $I_a$ be the excenter of $ABC$ opposite to $A$ , and let $M$ be the midpoint of $DI_a$ . Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$ .

Patrik Bak (Slovakia)