DE-MO 1996 Problem 5

Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$ . Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$ . Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.