AT-OEMO-FED1 2020 Problem 2

Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$ . On the sides $AC$ and $BC$ , the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$ . Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$ , respectively. Prove that $FG$ is half as long as $AB$ .

(Walther Janous)