USAJMO 2023 Problem 6

Isosceles triangle $ABC$ , with $AB=AC$ , is inscribed in circle $\omega$ . Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$ . Ray $AD$ intersects $\omega$ again at $E$ (other than $A$ ). Point $F$ (other than $E$ ) is chosen on $\omega$ such that $\angle DFE = 90^\circ$ . Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$ , respectively. Prove that $\angle XDE = \angle EDY$ .

Proposed by Anton Trygub