en>Phil Boswell: expand references using {{cite doi}}

2012-07-12T10:19:40Z

expand references using {{cite doi}}

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[[File:De_gua_theorem_1.svg|thumb|upright=1.4|tetrahedron with a right-angle corner in O]]
'''De Gua's theorem''' is a three-dimensional analog of the [[Pythagorean theorem]] and named for [[Jean Paul de Gua de Malves]].

If a [[tetrahedron]] has a right-angle corner (like the corner of a [[cube (geometry)|cube]]), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces.

:<math> A_{ABC}^2 = A_{\color {blue} ABO}^2+A_{\color {green} ACO}^2+A_{\color {red} BCO}^2 </math>

The [[Pythagorean theorem]] and de Gua's theorem are special cases (''n'' = 2, 3) of a [[simplex#Simplexes with an "orthogonal corner"|general theorem]] about [[simplex|''n''-simplices]] with a [[right angle]] corner.

Jean Paul de Gua de Malves (1713–85) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, ''Tinseau d'Amondans'' (1746–1818), as well. However the theorem had been known much earlier to [[Johann Faulhaber]] (1580–1635) and [[René Descartes]] (1596–1650).<ref>{{MathWorld|title=de Gua's theorem|urlname=deGuasTheorem}}</ref><ref>Hans-Bert Knoop: ''Ausgewählte Kapitel zur Geschichte der Mathematik''. Lecture Notes (University of Düsseldorf), p. 55 ([http://www.uni-due.de/~hn213me/mt/s08/geschichte/par04.pdf ''§ 4 Pythagoreische n-Tupel'', p. 50-65]) (German)</ref>

==Notes==
<References/>

==References==
* {{MathWorld|title=de Gua's theorem|urlname=deGuasTheorem}}
* Sergio A. Alvarez: [http://www.cs.bc.edu/~alvarez/NDPyt.pdf ''Note on an n-dimensional Pythagorean theorem''], Carnegie Mellon University.
*[http://www.gogeometry.com/solid/gua_theorem.htm ''De Gua's Theorem, Pythagorean theorem in 3-D''] — Graphical illustration and related properties of the tetrahedron.

== Further reading ==
* {{cite journal |last1=Kheyfits |first1=Alexander |year=2004 |title=The Theorem of Cosines for Pyramids |journal=The College Mathematics Journal |publisher=Mathematical Association of America |volume=35 |issue=5 |pages=385–388 |jstor=4146849}} Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.

{{DEFAULTSORT:De Gua'S Theorem}}
[[Category:Theorems in geometry]]
[[Category:Euclidean geometry]]

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en>Phil Boswell: expand references using {{cite doi}}