Midpoints of a quadrilateral

Join the midpoints P, Q, R, S of the sides AB, BC, CD, DA of any quadrilateral: PQRS is always a parallelogram. This is Varignon's theorem.

Why: the Midpoint Theorem, used on each diagonal:

• in ΔABC, PQ ‖ AC and PQ = AC/2;
• in ΔADC, SR ‖ AC and SR = AC/2;
• in the same way, QR and PS are parallel to BD and half of it.

So the sides of PQRS are half the diagonals of ABCD, and its perimeter is AC + BD.

Special cases:

• AC = BD ⇒ PQRS is a rhombus;
• AC ⊥ BD ⇒ PQRS is a rectangle;
• both ⇒ a square.

It works for every 4-gon. The proof never uses convexity, so it is true for dented, self-intersecting and even non-planar 4-gons.

The converse fails. Points on the sides can form a parallelogram without being midpoints: take points at equal fractions round a square.