AustriaMO2011

Let $ABC$ be an isosceles triangle with $\overline{AC}=\overline{BC}$ and $P$ be a point of the circumcircle lying on the arc $CA$ not containing $B$. Let $E$ and $F$ be the orthogonal projections of the point $C$ onto the lines $AP$ and $BP$, respectively. Prove that $AE$ and $BF$ have the same length. W. Janous, Innsbruck