Question 125
A circle with center $I$ touches the sides $AB$, $BC$, $AC$ of non-isosceles triangle $ABC$ at points $C_1$, $A_1$, $B_1$, respectively. Circles $\omega_B$ and $\omega_C$ are inscribed into the quadrilaterals $BA_1IC_1$ and $CA_1IB_1$, respectively. Prove that the common internal tangent to $\omega_B$ and $\omega_C$, distinct from line $IA_1$, passes through $A$. (I. Bogdanov)