De Gua's theorem
In
Generalizations
The
Let U be a measurable subset of a k-dimensional affine subspace of (so ). 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
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 with vertices on the co-ordinate axes. For example, suppose n = 3, k = 2 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
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.[4][5]
History
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, 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).[6][7]
See also
Notes
- S2CID 224956341.
- S2CID 125391795.
- JSTOR 2319528.
- JSTOR 4146849.
- ISSN 0343-6993.
- ^ Weisstein, Eric W. "de Gua's theorem". MathWorld.
- )
References
- Sergio A. Alvarez: Note on an n-dimensional Pythagorean theorem, Carnegie Mellon University.
- Hull, Lewis; Perfect, Hazel; Heading, J. (1978). "62.23 Pythagoras in Higher Dimensions: Three Approaches". Mathematical Gazette. 62 (421): 206–211. S2CID 187356402.
- Weisstein, Eric W. "de Gua's theorem". MathWorld.