PL-OM 2021 Problem 3

Let $\omega$ be the circumcircle of a triangle $ABC$ . Let $P$ be any point on $\omega$ different than the verticies of the triangle. Line $AP$ intersects $BC$ at $D$ , $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$ . Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$ , $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$ . Let $Q$ be the circumcenter of triangle $XYZ$ . Prove that all possible points $Q$ , corresponding to different positions of $P$ lie on one circle.