De Gua's theorem
In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner, 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:
De Gua's theorem can be applied for proving a special case of Heron's formula.
Generalizations
The Pythagorean theorem and de Gua's theorem are special cases of a general theorem about n-simplices with a right-angle corner, proved by P. S. Donchian and H. S. M. Coxeter in 1935. 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 . For any subset with exactly k elements, let be the orthogonal projection of U onto the linear span of, where and is the standard basis for. Then
where is the k-dimensional volume of U and the sum is over all subsets with exactly k elements.
De Gua's theorem and its generalisation to n-simplices with right-angle corners correspond to the special case where k = n−1 and U is an -simplex in with vertices on the co-ordinate axes. For example, suppose, and U is the triangle in with vertices A, B and C lying on the -, - and -axes, respectively. The subsets of with exactly 2 elements are, and. By definition, is the orthogonal projection of onto the -plane, so is the triangle with vertices O, B and C, where O is the origin of. Similarly, and, so the Conant–Beyer theorem says
which is de Gua's theorem.
The generalisation of de Gua's theorem to n-simplices with right-angle corners can also be obtained as a special case from the Cayley–Menger determinant formula.
De Gua's theorem can also be generalized to arbitrary tetrahedra and to pyramids, similarly to how the law of cosines generalises Pythagoras' theorem.