GeoProof — Parallelogram Law explorer

AC² + BD² = 2(a² + b²)

Parallelogram Law

In any parallelogram, the two diagonals and the two sides are tied together by one identity:

AC² + BD² = 2(a² + b²)

The sum of the squares of the diagonals equals twice the sum of the squares of a pair of adjacent sides — no matter how you skew or stretch the shape.

Where it comes from

Write the sides as vectors u and v. The diagonals are u + v and u − v. Then |u+v|² + |u−v|² = 2|u|² + 2|v|², because the cross terms (±2u·v) cancel. That cancellation is the whole theorem.

The rectangle case

In a rectangle the diagonals are equal, so AC² + BD² = 2·AC², giving AC² = a² + b² — the Pythagorean theorem. Use the snap button to see it.

Why it matters

It characterizes "Euclidean" length itself: a way of measuring distance comes from a true inner product exactly when it obeys this law. It's foundational in linear algebra and analysis.