OME 2007 Problem 6

Given a halfcircle of diameter $AB = 2R$ , consider a chord $CD$ of length $c$ . Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$ . Prove that the segment $EF$ has a constant length and direction when varying the chord $CD$ about the halfcircle.