BG-MO 2009 Problem 2

In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$ . The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$ . Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.