Prove that the quadrilateral PQRS, whose vertices are the midpoints of the sides of an arbitrary quadrilateral ABCD, is a parallelogram. This is an exercise in a linear algebra textbook so I would like to solve it using vectors.
I tried expressing the sides of the parallelogram in terms of the half-sizes of $ABCD$ eg. $PQ = PB + BQ$ but I that probably isn't enough information because the resulting equations did not lead anywhere useful. (Relevant figure can be seen here).