ELMO SL 2014 G7

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$ , let $\omega_A$ be its $A$ -mixtilinear incircle, $\omega_B$ be its $B$ -mixtilinear incircle, $\omega_C$ be its $C$ -mixtilinear incircle, and $X$ be the radical center of $\omega_A$ , $\omega_B$ , $\omega_C$ . Let $A'$ , $B'$ , $C'$ be the points at which $\omega_A$ , $\omega_B$ , $\omega_C$ are tangent to $\omega$ . Prove that $AA'$ , $BB'$ , $CC'$ and $OX$ are concurrent.

Proposed by Robin Park