RMM 2012 Problem 6

Let $A B C$ be a triangle and let $I$ and $O$ respectively denote its incentre and circumcentre. Let $\omega_{A}$ be the circle through $B$ and $C$ and tangent to the incircle of the triangle $A B C$; the circles $\omega_{B}$ and $\omega_{C}$ are defined similarly. The circles $\omega_{B}$ and $\omega_{C}$ through $A$ meet again at $A^{\prime}$; the points $B^{\prime}$ and $C^{\prime}$ are defined similarly. Prove that the lines $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ are concurrent at a point on the line $I O$.