Question 1
Let $ABCD$ be a quadrilateral with $AB=BC=CD=DA$. Let $MN$ and $PQ$ be segments perpendicular to diagonal $BD$, at distance $d\gt BD/2$ from each other, where $M\in AD$, $N\in DC$, $P\in AB$, $Q\in BC$. Show that the perimeter of hexagon $AMNCQP$ is independent of the positions of $MN,PQ$ provided their distance remains constant.
