RMO 2023 Problem 2

Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the arc $A B$ . Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$ . Prove that (a) the measure of $\angle C E D$ is a constant; (b) the circumcircle of triangle $C E D$ passes through a fixed point.