GeoProof — Varignon's Theorem explorer

PQ = RS = ½·AC · QR = SP = ½·BD

Drag any corner — convex, concave, or crossed — and the inner shape stays a perfect parallelogram.

Varignon's Theorem

Take any quadrilateral ABCD and mark the midpoint of each side: P on AB, Q on BC, R on CD, S on DA. Connect them in order. The result is always a parallelogram — the Varignon parallelogram:

PQ ∥ RS, QR ∥ SP

This works for every quadrilateral: convex, concave, even a self-crossing one.

Why it's true

Look at triangle ABC: P and Q are midpoints of two of its sides, so segment PQ is a midsegment — parallel to the third side AC and exactly half its length. In triangle ACD, the segment SR is also a midsegment of the same diagonal AC, so SR is parallel to AC and half its length too. Therefore PQ and SR are parallel and equal. The same argument with diagonal BD handles the other pair.

Sides, area, and perimeter

Each pair of parallelogram sides equals half a diagonal of ABCD, so its perimeter equals AC + BD (the sum of the diagonals). For a simple quadrilateral, its area is exactly half the area of ABCD.

A neat special case

If the diagonals of ABCD are equal (as in an isosceles trapezoid), all four parallelogram sides are equal — a rhombus. If the diagonals are perpendicular (as in a kite), the parallelogram is a rectangle.

A bit of history

It is named for Pierre Varignon (1654–1722), the French mathematician, and was published in his work on mechanics and geometry in 1731.