it can be easily proved using euclidean geometry rather that vectors.
consider two parallel planes plane 1 and plane 2. ABCD is a parallelogram in plane 1 and PQRS is a parallelogram in plane 2. these two parallelograms are such that AP || BQ || CR || DS.
now ABCDPQRS is an arbitrary parallelopiped.
to prove: AR bisects BS.
observe that A,B,R,S are coplanar.
also: AS=BR and AB=RS.
so ABRS is a parallogram. since we know that diagonals of a parallelogram bisect each other we have AR bisects BS.