De Gua's theorem

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 by Donald R. Conant and William A. Beyer, which can be stated as follows.

Let U be a measurable subset of a k-dimensional affine subspace of R n (so k ≤ n ). For any subset I ⊆ { 1 , … , n } with exactly k elements, let U I be the orthogonal projection of U onto the linear span of e i 1 , … , e i k , where I = { i 1 , … , i k } and e 1 , … , e n is the standard basis for R n . Then

where vol k ( U ) is the k-dimensional volume of U and the sum is over all subsets I ⊆ { 1 , … , n } with exactly k elements.

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 U is an (n−1)-simplex in R n with vertices on the co-ordinate axes. For example, suppose n = 3, k = 2 and U is the triangle △ A B C in R 3 with vertices A, B and C lying on the x 1 -, x 2 - and x 3 -axes, respectively. The subsets I of { 1 , 2 , 3 } with exactly 2 elements are { 2 , 3 } , { 1 , 3 } and { 1 , 2 } . By definition, U { 2 , 3 } is the orthogonal projection of U = △ A B C onto the x 2 x 3 -plane, so U { 2 , 3 } is the triangle △ O B C with vertices O, B and C, where O is the origin of R 3 . Similarly, U { 1 , 3 } = △ A O C and U { 1 , 2 } = △ A B O , so the Conant–Beyer theorem says

which is de Gua's theorem.

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).