GeoProof — British Flag Theorem explorer

PA² + PC² = PB² + PD²

Drag P anywhere — even outside the rectangle — and the two totals stay equal. Drag the top-right corner to resize.

The British Flag Theorem

Take any rectangle with corners A, B, C, D (in order, so A and C are opposite, and B and D are opposite). Pick any point P at all — inside the rectangle, on an edge, or far outside. Then the sums of the squared distances to each pair of opposite corners are equal:

PA² + PC² = PB² + PD²

The name comes from the way the four lines from an interior P to the corners resemble the diagonals on the Union Jack flag.

Why it's true

Drop P's coordinates relative to the rectangle's sides. Using the Pythagorean theorem on the horizontal and vertical gaps, each squared distance splits into a horizontal part and a vertical part. When you add the two opposite corners, the cross terms cancel and both sums collapse to the same total — twice the squared horizontal offset plus twice the squared vertical offset, plus the rectangle's own half-dimensions. So the two diagonal pairs must match.

It works outside too

Nothing in the argument assumes P is inside. Drag P well beyond a corner and the equality still holds exactly — the badge tells you whether P is currently inside or outside.

Needs a rectangle

The right angles are essential. The identity generally fails for a non-rectangular quadrilateral; it's the rectangle's equal, perpendicular sides that make the cross terms cancel.

A 3-D cousin

The same idea extends to a box (rectangular cuboid): for opposite corners of the box, the sums of squared distances from any point are again equal.

A bit of history

It's a modern teaching favorite rather than a classical named result — a direct, memorable consequence of the Pythagorean theorem applied twice.