De Gua's theorem

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), 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.

$A_{ABC}^2 = A_{\color {blue} ABO}^2+A_{\color {green} ACO}^2+A_{\color {red} BCO}^2$

The Pythagorean theorem and de Gua's theorem are special cases (n = 2, 3) of a general theorem about n-simplices with a right angle corner.

Jean Paul de Gua de Malves (1713-1785) 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).^[1][2]

Notes

1. ^ Weisstein, Eric W., "de Gua's theorem" from MathWorld.
2. ^ Hans-Bert Knoop: Ausgewählte Kapitel zur Geschichte der Mathematik. Lecture Notes (University of Düsseldorf), p. 55 (§ 4 Pythagoreische n-Tupel, p. 50-65) (German)

References

• Weisstein, Eric W., "de Gua's theorem" from MathWorld.
• Sergio A. Alvarez: Note on an n-dimensional Pythagorean theorem, Carnegie Mellon University
• De Gua's Theorem, Pythagorean theorem in 3-D - graphical illustration and related properties of the tetrahedron

Further reading

• Kheyfits, Alexander (2004). "The Theorem of Cosines for Pyramids". The College Mathematics Journal (Mathematical Association of America) 35 (5): 385–388. JSTOR 4146849.  Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.