British flag theorem

For any point in a rectangle, the sum of the squared distances to one pair of opposite vertices equals the sum for the other pair.

Coordinate geometry: The use of coordinates and algebra to solve geometric problems. Placing the rectangle at the origin reduces the theorem to cancellation of coordinate terms.
Midpoint: The point halfway between two points on a line segment. The rectangle’s center is the common midpoint of both diagonals.
Distance geometry: The study of geometric configurations through distances between points. The theorem is a fixed relation among four distances in a planar configuration.
William Rowan Hamilton: An Irish mathematician who made foundational contributions to algebra, mechanics, and optics. The theorem’s name is associated with Hamilton, though its attribution is not firmly established.
Pythagorean theorem: A relation stating that the square of a right triangle’s hypotenuse equals the sum of the squares of its legs. Each squared distance in the theorem follows from this relation.
Diagonal of a rectangle: A line segment joining opposite vertices of a rectangle. The theorem compares distances to the endpoints of its two diagonals.
Trilateration: A method for locating a point from its distances to known reference points. Distance identities can check or simplify measurements involving rectangle vertices.
British Mathematical Olympiad: An annual mathematics competition for school students in the United Kingdom. The theorem is widely circulated as a concise olympiad-style geometry result.
Parallelogram law: An identity stating that the sum of squared diagonals of a parallelogram equals the sum of squared sides. Its vector form gives the same diagonal-sum identity for a rectangle.
Squared Euclidean distance: The sum of the squared coordinate differences between two points in Euclidean space. The theorem concerns sums of squared, rather than unsquared, distances.

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Euler's quadrilateral theoremRelated: It is another squared-distance identity arising from rectangle geometry.