Van Aubel's theorem

The theorem can be applied to non-overturned and overturned squares.

In plane geometry , Van Aubel's theorem describes a relationship between the squares constructed over the sides of a quadrangle . The theorem says that the two lines between the centers of opposing squares are of equal length and are perpendicular to each other. In other words: the centers of the four squares are the corners of an orthodiagonal square with diagonals of equal length. The sentence is named after Henri van Aubel (1830-1906), a mathematics teacher at the Atheneum (high school) in Antwerp , who published it in 1878.

The sentence also applies to the inwardly constructed squares on the sides of the square. It should be noted that the square does not have to be convex.

See also

• Napoleon triangle
• Napoleon point

literature

• Yutaka Nishiyama: The Beautiful Geometric Theorem of Van Aubel (PDF) In: International Journal of Pure and Applied Mathematics , Volume 66, N3. 1, 2011, pp. 71-80
• D. Pellegrinetti: The Six-Point Circle for the Quadrangle . In: International Journal of Geometry , Volume 8 (2019), No. 2, pp. 5-13

Web links

Commons : Van Aubel's sentence  - collection of images, videos and audio files

• Eric W. Weisstein : van Aubel's theorem . In: MathWorld (English).
• Jay Warendorff: Van Aubel's Theorem for Quadrilaterals and Van Aubel's Theorem for Triangles - The Wolfram Demonstrations Project .
• Van Aubel's Theorem and some generalizations , an interactive dynamic geometry sketch at Dynamic Geometry Sketches

Individual evidence

1. ^ HH van Aubel: Note concernant les centers de carrés construits sur les côtés d'un polygon quelconque . In: Nouvelle Correspondance Mathématique . tape 4 , 1878, p. 40-44 (French).
2. ^ David Wells: Penguin Dictionary of Curious and Interesting Geometry . Penguin 1991, p. 11, there as Aubel's theorem