APMO 1996 Problem 1

Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$ . Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$ , with $M \in AD$ , $N \in DC$ , $P \in AB$ , and $Q \in BC$ . Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.