Problem

Source: Iran MO 2019, secound round, day 1, p2

Tags: geometry



$ABC$ is an isosceles triangle ($AB=AC$). Point $X$ is an arbitrary point on $BC$. $Z \in AC$ and $Y \in AB$ such that $\angle BXY = \angle ZXC$. A line parallel to $YZ$ passes through $B$ and cuts $XZ$ at $T$. Prove that $AT$ bisects $\angle A$.