KöMaL 2014 C1208

The vertices of the parallelograms PQRL and LSTA in the plane are labelled in the same sense around the clock. The parallelograms do not have a point in common, except $L$. Prove that there exists a pentagon ABCDE (degenerated cases are allowed) in the plane such that the midpoints of the sides are $P$, $Q$, $R$, $S$, $T$, in this order.