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van Aubel's Theorem

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AubelsTheorem

Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two lines are of equal length and cross at a right angle.

van Aubel's theorem is related to Napoleon's theorem and is a special case of the Petr-Neumann-Douglas theorem. It is sometimes (incorrectly) known simply as Aubel's theorem (Casey 1888; Wells 1991, p. 11; Kimberling 2003, p. 23).

A second theorem sometimes known as van Aubel's theorem states that if DeltaA^'B^'C^' is the Cevian triangle of a point P, then

(AP)/(PA^')=(AB^')/(B^'C)+(AC^')/(C^'B).

See also

Kiepert Hyperbola, Napoleon's Theorem, Petr-Neumann-Douglas Theorem, Quadrilateral, Right Angle, Square

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.de Villiers, M. D. "Dual Generalizations of Van Aubel's Theorem." Math. Gaz., 82, 405-412, 1998.de Villiers, M. D. "More on Dual Van Aubel Generalizations." Math. Gaz. 84, 121-122, 2000.de Villiers, M. D. "Generalizing Van Aubel Using Duality." Math. Mag. 73, 303-307, 2000.Kimberling, C. Geometry in Action: A Discovery Approach Using the Geometer's Sketchpad. Key Curriculum Press, p. 23, 2003.Kitchen, E. "Dörrie Tiles and Related Miniatures." Math. Mag. 67, 128-130, 1994.Kontogiannis, D. G. Equalities and Inequalities in the Triangle. Athens: Ekpaideutikis, p. 124, 1996.Silvester, J. R. "Extensions of a Theorem of Van Aubel." Math. Gaz. 90, 2-12, 2006.van Aubel, H. H. "Note concernant les centres de carrés construits sur les côtés d'un polygon quelconque." Nouv. Corresp. Math. 4, 40-44, 1878.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 11, 1991.Yaglom, I. M. Geometric Transformations I. New York: Random House, pp. 95-96, 1962.

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van Aubel's Theorem

Cite this as:

Weisstein, Eric W. "van Aubel's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/vanAubelsTheorem.html

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