SHARYGIN 2016 Problem 17

Let $D$ be an arbitrary point on side $BC$ of triangle $ABC$ . Circles $\omega_1$ and $\omega_2$ pass through $A$ and $D$ in such a way that $BA$ touches $\omega_1$ and $CA$ touches $\omega_2$ . Let $BX$ be the second tangent from $B$ to $\omega_1$ , and $CY$ be the second tangent from $C$ to $\omega_2$ . Prove that the circumcircle of triangle $XDY$ touches $BC$ .