Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.
Problem
Source: Latvian TST for Baltic Way 2019, Problem 12
Tags: geometry, perimeter, geometric transformation, reflection, circumcircle