CONO-SUR 2017 Problem 4

Let $ABC$ an acute triangle with circumcenter $O$ . Points $X$ and $Y$ are chosen such that:

• $\angle XAB = \angle YCB = 90^\circ$ [/*]
• $\angle ABC = \angle BXA = \angle BYC$ [/*]
• $X$ and $C$ are in different half-planes with respect to $AB$ [/*]
• $Y$ and $A$ are in different half-planes with respect to $BC$ [/*]

Prove that $O$ is the midpoint of $XY$ .