Let $ABCD$ be any quadrilateral in the $xy$-plane.

Let $P,Q,R,S$ be the midpoints of $AB,BC,CD,DA$ respectively.

This diagram suggests that $PQRS$ is always a parallelogram.

Prove this is the case.

Find $P$ in terms of $A(x_A,y_A)$ and $B(x_B,y_B)$ using the midpoint formula, and do similar for $Q,R,S.$ Show algebraically that

and similar for $QR$ and $SP.$