IT-CESENATICO 2023 Problem 4

Fix circle with center $O$ , diameter $AB$ and a point $C$ on it, different from $A, B$ . Let a point $D$ , different from $A, B$ , vary on the arc $AB$ not containing $C$ . Let $E$ lie on $CD$ such that $BE \perp CD$ . Prove that $CE \cdot ED$ is maximal exactly when $BOED$ is cyclic.