CAPS 2022 Problem 5

Let $ABC$ be a triangle with $AB < AC$ and circumcenter $O$ . The angle bisector of $\angle BAC$ meets the side $BC$ at $D$ . The line through $D$ perpendicular to $BC$ meets the segment $AO$ at $X$ . Furthermore, let $Y$ be the midpoint of segment $AD$ . Prove that points $B, C, X, Y$ are concyclic.