RMM 2011 Problem 3

A triangle $A B C$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $B C$ meets segments $A B$, $A C$ at points $D, E$ respectively, and meets $\omega$ at points $K, L$ (where $D$ lies between $K$ and $E$ ). Circle $\gamma_{1}$ is tangent to the segments $K D$ and $B D$ and also tangent to $\omega$, while circle $\gamma_{2}$ is tangent to the segments $L E$ and $C E$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_{1}$ and $\gamma_{2}$. (Russia) VASily Mokin & Fedor IvLev