SHARYGIN 2019 Problem 22

Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$ . A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$ . Let $X$ be an arbitrary point of line $BC$ . Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$