De Gua's theorem

File:De gua theorem 1.svg

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.

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

Generalizations

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. This, in turn, is a special case of a yet more general theorem, which can be stated as follows.

Let P be a k-dimensional affine subspace of $\mathbb{R}^n$ (so $k \le n$) and let C be a compact subset of P. For any subset $I \subseteq \{ 1, \ldots, n \}$ with exactly k elements, let $C_I$ be the orthogonal projection of C onto the linear span of $e_{i_1}, \ldots, e_{i_k}$, where $I = \{i_1, \ldots, i_k\}$ and $e_1, \ldots, e_n$ is the standard basis for $\mathbb{R}^n$. Then

$\mbox{vol}_k^2(C) = \sum_I \mbox{vol}_k^2(C_I),$

where $\mbox{vol}_k(C)$ is the k-dimensional volume of C and the sum is over all subsets $I \subseteq \{ 1, \ldots, n \}$ with exactly k elements.

This theorem is essentially the inner-product-space version of Pythagoras’ theorem applied to the k^th exterior power of n-dimensional Euclidean space. De Gua's theorem and its generalisation (above) to n-simplices with right-angle corners correspond to the special case where k = n−1 and C is an (n−1)-simplex in $\mathbb{R}^n$ with vertices on the co-ordinate axes.

History

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, Charles de Tinseau d'Amondans (1746–1818), as well. However the theorem had also been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650).

References

• Weisstein, Eric W., "de Gua's theorem", 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

• Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.