Finsler–Hadwiger theorem

The Finsler–Hadwiger theorem states that two squares sharing a vertex determine a third square whose vertices include their centers and the vertices opposite the shared vertex.

Parallelogram law: In a Euclidean inner-product space, the sum of squared norms of two vector sums equals twice the sum of their squared norms. The identity supplies the equal-length relation behind the theorem’s third square.
Square center: The center of a square is the intersection of its diagonals and is equidistant from all four vertices. Each original square contributes its center as a vertex of the third.
Square tiling: A square tiling covers a plane or region with squares whose interiors do not overlap. Adjacent squares provide a simple visual arrangement for testing the vertex configuration.
Paul Finsler: Paul Finsler was a Swiss mathematician who worked in geometry and set theory. The theorem bears Finsler’s name alongside Hadwiger’s.
Complex numbers in geometry: Complex coordinates encode planar points and transformations using algebraic operations. Complex coordinates turn the square conditions into concise equations.
Diagonal of a square: A diagonal of a square joins opposite vertices and bisects its angles and the other diagonal. The theorem uses the vertices opposite the shared vertex.
Unit square: A unit square is a square whose side length is one. Normalizing one square’s side makes coordinate verification especially simple.
Hugo Hadwiger: Hugo Hadwiger was a Swiss mathematician known for work in geometry and convexity. Hadwiger’s name is paired with Finsler in the theorem’s standard title.
Square: A square is a quadrilateral with four equal sides and four right angles. The theorem follows from constraints imposed by three mutually related squares.
Perpendicularity: Two lines are perpendicular when they meet at a right angle. The resulting equal sides must also meet at right angles.

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Thébault's theoremRelated: It offers another center-based result arising from externally constructed regular figures.