Van Aubel's theorem

Van Aubel's theorem states that, for squares constructed externally on the sides of a quadrilateral, the segments joining the centers of opposite squares are equal in length and perpendicular.

Complex numbers in geometry: The use of complex arithmetic to represent points, transformations, and relationships in the plane. Representing square centers as complex points turns the theorem into an algebraic identity.
Quadrilateral: A polygon with four sides and four vertices. The theorem applies to squares constructed on the sides of this four-sided figure.
Rectangle: A quadrilateral with four right angles. A rectangle provides a familiar special case where the opposite-square center segments can be checked directly.
Henry Van Aubel: A Belgian mathematician associated with the theorem concerning squares constructed on a quadrilateral's sides. His name identifies the result, though detailed biographical information is limited.
Vector addition: The operation that combines directed quantities by adding their corresponding components. The theorem's perpendicular equal segments follow from cancellations among side vectors.
Square: A quadrilateral with four equal sides and four right angles. Its equal sides and quarter-turn geometry determine the constructed centers.
Parallelogram: A quadrilateral whose opposite sides are parallel. Its paired side vectors make the center relationships especially transparent.
Euclidean geometry: The geometry of points, lines, angles, and figures governed by Euclid's postulates. The theorem is a classical plane-geometric result built from squares and perpendicularity.
Rotation (mathematics): A transformation that turns points around a fixed center through a specified angle. Constructing a square uses a quarter-turn of each side vector.
Centroid: The geometric center of a plane figure, defined as the average position of its area. For each constructed square, the relevant center is also its center of symmetry.

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