# Prove that the quadrilateral whose vertices are the midpoints of the sides of an arbitrary quadrilateral is a parallelogram

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).

The midpoint of $AB$ is $\frac12(A+B)$.
The midpoint of $BC$ is $\frac12(B+C)$.
The vector from the midpoint of $AB$ to the midpoint of $BC$ is $\frac12(B+C)-\frac12(A+B) = \frac12(C-A)$.
Repeat for $AD$ and $DC$ to see that you get the same vector there.