Euler's quadrilateral theorem

For any quadrilateral, the sum of the squares of its four side lengths equals the sum of the squares of its diagonal lengths plus four times the squared distance between the diagonal midpoints.

Parallelogram law: A vector identity equating the sum of two vectors’ squared lengths and the sum of the squared lengths of their sum and difference. Applying this identity to paired sides produces the theorem’s diagonal and midpoint terms.
Squared Euclidean distance: The sum of the squares of coordinate differences between two points in Euclidean space. Every length in the theorem is squared, allowing coordinate and vector proofs.
Varignon's theorem: The theorem that joining the midpoints of a quadrilateral’s sides produces a parallelogram. Its midpoint geometry provides a related way to study lengths inside any quadrilateral.
Euler's theorem for triangles: A relation between a triangle’s side lengths, medians, and the lengths of its medians. It is a distinct Euler identity for triangles rather than the four-vertex midpoint formula.
Vector (geometric): A quantity with magnitude and direction, represented geometrically by a directed segment. Representing vertices by position vectors turns the theorem into an algebraic identity.
Convex quadrilateral: A quadrilateral whose interior angles are all less than 180 degrees. Convex examples make the relation between the four sides and crossing diagonals easy to visualize.
British flag theorem: A theorem stating that, for a rectangle, the sum of squared distances from an interior point to opposite vertices is equal for each pair. It is another squared-distance identity arising from rectangle geometry.
Parallelogram law in inner product spaces: An identity stating that the sum of squared norms of two vectors and their sum and difference obey a fixed equality. Unlike the quadrilateral theorem, it concerns two vectors and has no independent midpoint correction.
Midpoint: The point halfway between the endpoints of a segment. The correction term is four times the squared distance between the two diagonal midpoints.
Rectangle: A quadrilateral with four right angles. Its diagonals bisect one another, so the midpoint correction vanishes.

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