RIOPLATENSE 2015 Problem 6

Let $A B C$ be an acut-angles triangle of incenter $I$ , circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$ . $A_1$ is the point of $\omega$ such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$ . Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$ , tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$ , is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$ . Similarly are defined points $B_2$ and $C_2$ . Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.