Van Aubel's theorem

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The theorem can be applied to a complex (self-intersecting) quadrilateral. Van-Aubel-theorem combined.svg
The theorem can be applied to a complex (self-intersecting) quadrilateral.

In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of an equidiagonal orthodiagonal quadrilateral. The theorem is named after H. H. van Aubel, who published it in 1878. [1]

Contents

The theorem holds true also for re-entrant quadrilaterals, [2] and when the squares are constructed internal to the given quadrilateral. [3] For complex (self-intersecting) quadrilaterals, the external and internal constructions for the squares are not definable. In this case, the theorem holds true when the constructions are carried out in the more general way: [3]

A few extensions of the theorem, considering similar rectangles, similar rhombi and similar parallelograms constructed on the sides of the given quadrilateral, have been published on the The Mathematical Gazette. [4] [5]

See also

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References

  1. van Aubel, H. H. (1878), "Note concernant les centres de carrés construits sur les côtés d'un polygon quelconque", Nouvelle Correspondance Mathématique (in French), 4: 40–44.
  2. Coxeter, H.S.M., and Greitzer, Samuel L. 1967. Geometry Revisited, pages 52.
  3. 1 2 D. Pellegrinetti: "The Six-Point Circle for the Quadrangle". International Journal of Geometry, Vol. 8 (Oct., 2019), No. 2, pp. 5–13.
  4. M. de Villiers: "Dual Generalizations of Van Aubel's theorem". The Mathematical Gazette, Vol. 82 (Nov., 1998), pp. 405-412.
  5. J. R. Silvester: "Extensions of a Theorem of Van Aubel". The Mathematical Gazette, Vol. 90 (Mar., 2006), pp. 2-12.