AT-OEMO-BEG 2011 Problem 4

Let $ABC$ be an isosceles triangle with $AC = BC$ . On the arc $CA$ of its circumcircle, which does not contain $B$ , there is a point $P$ . The projection of $C$ on the line $AP$ is denoted by $E$ , the projection of $C$ on the line $BP$ is denoted by $F$ . Prove that the lines $AE$ and $BF$ have equal lengths.

(W. Janous, WRG Ursulincn, Innsbruck)