OMM 2019 Problem 6

Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$ . Let $H,O$ be the orthocenter and circumcenter of $ABC$ , respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$ . Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$ , respectively.