De Gua's theorem

De Gua's theorem Tetrahedron with a right-angle corner in O 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 (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: {estilo de exibição A_{abc}^{2}=A_{cor {azul}ABO}^{2}+UMA_{cor {verde}ACO}^{2}+UMA_{cor {vermelho}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, por sua vez, is a special case of a yet more general theorem by Donald R. Conant and William A. Beyer,[1] which can be stated as follows.

Let U be a measurable subset of a k-dimensional affine subspace of {estilo de exibição mathbb {R} ^{n}} (assim {displaystyle kleq n} ). For any subset {displaystyle Isubseteq {1,ldots ,n}} with exactly k elements, deixar {estilo de exibição U_{EU}} be the orthogonal projection of U onto the linear span of {displaystyle e_{eu_{1}},ldots ,e_{eu_{k}}} , Onde {estilo de exibição I={eu_{1},ldots ,eu_{k}}} e {displaystyle e_{1},ldots ,e_{n}} is the standard basis for {estilo de exibição mathbb {R} ^{n}} . Então {nome do operador de estilo de exibição {volume} _{k}^{2}(você)=soma _{EU}nome do operador {volume} _{k}^{2}(VOCÊ_{EU}),} Onde {nome do operador de estilo de exibição {volume} _{k}(você)} is the k-dimensional volume of U and the sum is over all subsets {displaystyle Isubseteq {1,ldots ,n}} with exactly k elements.

De Gua's theorem and its generalisation (acima de) to n-simplices with right-angle corners correspond to the special case where k = n−1 and U is an (n−1)-simplex in {estilo de exibição mathbb {R} ^{n}} with vertices on the co-ordinate axes. Por exemplo, suppose n = 3, k = 2 and U is the triangle {displaystyle triangle ABC} dentro {estilo de exibição mathbb {R} ^{3}} with vertices A, B and C lying on the {estilo de exibição x_{1}} -, {estilo de exibição x_{2}} - e {estilo de exibição x_{3}} -axes, respectivamente. The subsets {estilo de exibição I} do {estilo de exibição {1,2,3}} with exactly 2 elements are {estilo de exibição {2,3}} , {estilo de exibição {1,3}} e {estilo de exibição {1,2}} . Por definição, {estilo de exibição U_{{2,3}}} is the orthogonal projection of {displaystyle U=triangle ABC} onto the {estilo de exibição x_{2}x_{3}} -plane, assim {estilo de exibição U_{{2,3}}} is the triangle {displaystyle triangle OBC} with vertices O, B and C, where O is the origin of {estilo de exibição mathbb {R} ^{3}} . De forma similar, {estilo de exibição U_{{1,3}}=triangle AOC} e {estilo de exibição U_{{1,2}}=triangle ABO} , so the Conant–Beyer theorem says {nome do operador de estilo de exibição {volume} _{2}^{2}(triangle ABC)=nome do operador {volume} _{2}^{2}(triangle OBC)+nome do operador {volume} _{2}^{2}(triangle AOC)+nome do operador {volume} _{2}^{2}(triangle ABO),} 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.

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), também. However the theorem had also been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650).[2][3] See also Vector area and projected area Bivector Notes ^ Donald R Conant & William A Beyer (Mar 1974). "Generalized Pythagorean Theorem". O American Mathematical Monthly. Associação Matemática da América. 81 (3): 262–265. doi:10.2307/2319528. JSTOR 2319528. ^ Weisstein, Eric W. "de Gua's theorem". MathWorld. ^ Howard Whitley Eves: Great Moments in Mathematics (antes 1650). Associação Matemática da América, 1983, ISBN 9780883853108, S. 37 (excerpt, p. 37, at Google Books) 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 Kheyfits, Alexandre (2004). "The Theorem of Cosines for Pyramids". O Jornal de Matemática da Faculdade. Associação Matemática da América. 35 (5): 385-388. JSTOR 4146849. Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids. Lévy-Leblond, Jean Marc (2020). "The Theorem of Cosines for Pyramids". The Mathematical Intelligencer. SpringerLink. Application of de Gua's theorem for proving a special case of Heron's formula. Categorias: Theorems in geometryEuclidean geometry