VMO 1965 Problem 2

$AB$ and $CD$ are two fixed parallel chords of the circle $S$ . $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$ . $X$ is the circumcenter of the triangle $MCQ$ . Find the locus of $X$ . What happens to $X$ as $M$ tends to (1) $D$ , (2) $C$ ? Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$ .