ELMO SL 2014 G8

In triangle $ABC$ with incenter $I$ and circumcenter $O$ , let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$ -mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$ , and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$ , and that $\angle AXO = \angle OXA'$ .

Proposed by Sammy Luo