De Gua's theorem

Tetrahedron with right-angled corner in O

The de Gua theorem is a spatial analogue of the Pythagorean theorem and named after Jean Paul de Gua de Malves (1713–1785), who published it in 1783.

If a tetrahedron has a right-angled corner (like a cube corner), then the sum of the squared areas of the areas adjacent to the right-angled corner is equal to the squared area of ​​the area opposite the right angle.

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

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

The sentence was published at the same time in a somewhat more general form by the French mathematician Tinseau d'Amondans (1746-1818) and was even known much earlier to René Descartes (1596-1650) and Johannes Faulhaber (1580-1635).

Web links

• Eric W. Weisstein : de Gua's theorem . In: MathWorld (English).
• Sergio A. Alvarez: Note on an n-dimensional Pythagorean theorem. (PDF; 71 kB) Carnegie Mellon University (English)
• De Gua's Theorem, Pythagorean theorem in 3-D . graphical representation and related properties

Individual evidence

1. ↑ Eric W. Weisstein : de Gua's Theorem . In: MathWorld (English).
2. ↑ Howard Whitley Eves: Great Moments in Mathematics (before 1650) . Mathematical Association of America, 1983, ISBN 978-0-88385-310-8 , p. 37 ( excerpt (Google) )