KöMaL 2007 B4007

Let $A$, $B$, $C$, $D$ denote four distinct points in the space. The line segment $PQ$ is ``moved'' from $AB$ to $DC$ so that in each time instant $P$ divides the line segment $AD$ in the same ratio as $Q$ divides $BC$. Then the line segment $PQ$ sweeps out a surface. Prove that the same surface will be obtained if a line segment $RS$ is moved in the same way from $AD$ to $BC$.