RU-ALLRUSSIAN 2022 Problem 113

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$ . Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$ , and the point $Q$ is on the smaller arc $AC$ . Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$ , from point $Q$ to the line $AC$ and from point $A$ to line $PQ$ . Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.