Thébault's theorem

Thébault's theorem states that squares constructed externally on the sides of a parallelogram have centers that form a square.

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Parallelogram: A quadrilateral whose opposite sides are parallel. Its parallel opposite sides supply the paired directions used in the square-center construction.
Center of a square: The point equidistant from all four vertices of a square. Each constructed square contributes this point to the four-vertex figure.
Napoleon's theorem: A theorem stating that equilateral triangles erected externally on a triangle have centers forming an equilateral triangle. It has a parallel construction using centers of externally erected regular polygons.
Victor Thébault: A French mathematician known for posing and solving problems in elementary geometry. His name is attached to this parallelogram and square-center result.
Square: A quadrilateral with four equal sides and four right angles. The theorem’s constructed squares and resulting center figure share this defining shape.
External angle bisector: A line that bisects the angle between the extensions of two intersecting lines. The square centers lie along directions determined by outward normals to the parallelogram’s sides.
Van Aubel's theorem: A theorem stating that squares erected on the sides of a quadrilateral yield perpendicular, equal segments between opposite square centers. It studies the same square-center construction but gives a different relation for quadrilaterals.
Paul Erdős: A Hungarian mathematician known for extensive collaborations and problems across mathematics. His name is associated with several related geometric problems involving erected squares.
Complex plane: A coordinate plane in which points are represented by complex numbers. Complex multiplication compactly represents the quarter-turns used to locate the square centers.
Perpendicular: Two lines that intersect at a right angle. The square side directions are perpendicular to the corresponding parallelogram sides.