RU-ALLRUSSIAN 2009 Problem 107

The incircle $(I)$ of a given scalene triangle $ABC$ touches its sides $BC$ , $CA$ , $AB$ at $A_1$ , $B_1$ , $C_1$ , respectively. Denote $\omega_B$ , $\omega_C$ the incircles of quadrilaterals $BA_1IC_1$ and $CA_1IB_1$ , respectively. Prove that the internal common tangent of $\omega_B$ and $\omega_C$ different from $IA_1$ passes through $A$ .