Midpoints of Sides of Quadrilateral form Parallelogram

Theorem
Let $\Box ABCD$ be a quadrilateral.

Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respectively.

Then $\Box EFGH$ is a parallelogram.

Proof
Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices of $\Box ABCD$.

The midpoints of the sides of $\Box ABCD$ are, then:

Take two opposite sides $EF$ and $HG$:

The vectors defining the opposite sides of $\Box EFGH$ are equal.

The result follows by definition of parallelogram.