OBM 1984 Problem 3

Given a regular dodecahedron of side $a$ . Take two pairs of opposite faces: $E, E'$ and $F, F'$ . For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.