IR-TST 2022 Problem 8

In triangle $ABC$ , with $AB<AC$ , $I$ is the incenter, $E$ is the intersection of $A$ -excircle and $BC$ . Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$ . Point $Q$ lies on $BC$ such that $\angle AIQ=90$ . Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$ , circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$ . if $M$ is the Midpoint od the arc $BC$ , which does not contain $A$ , prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$ .

Proposed by Amirmahdi Mohseni