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 $BC$.
Problem
Source: Sharygin Geometry Olympiad 2016 First Round P17 grades 9-11
Tags: geometry, circumcircle, tangent, circles