UA-UMO 2005 Problem 98

Let $AB$ and $CD$ be two disjoint chords of a circle. A point $E$ , distinct from $A$ and $B$ , is taken on the chord $AB$ . Consider the arc $AB$ not containing $C$ and $D$ . Using a ruler and a compass, construct a point $F$ on this arc such that $\frac{PE}{EQ}=\frac{1}{2}$ , where $P$ and $Q$ are the intersection points of $AB$ with the segments $FC$ and $FD$ , respectively.