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: {stile di visualizzazione A_{ABC}^{2}=A_{colore {blu}ABO}^{2}+UN_{colore {verde}ACO}^{2}+UN_{colore {rosso}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, a sua volta, 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 {displaystyle mathbb {R} ^{n}} (Così {displaystyle kleq n} ). For any subset {displaystyle Isubseteq {1,ldot ,n}} with exactly k elements, permettere {stile di visualizzazione U_{io}} be the orthogonal projection of U onto the linear span of {stile di visualizzazione e_{io_{1}},ldot ,e_{io_{K}}} , dove {stile di visualizzazione I={io_{1},ldot ,io_{K}}} e {stile di visualizzazione e_{1},ldot ,e_{n}} is the standard basis for {displaystyle mathbb {R} ^{n}} . Quindi {nome dell'operatore dello stile di visualizzazione {vol} _{K}^{2}(u)=somma _{io}nome operatore {vol} _{K}^{2}(U_{io}),} dove {nome dell'operatore dello stile di visualizzazione {vol} _{K}(u)} is the k-dimensional volume of U and the sum is over all subsets {displaystyle Isubseteq {1,ldot ,n}} with exactly k elements.

De Gua's theorem and its generalisation (sopra) to n-simplices with right-angle corners correspond to the special case where k = n−1 and U is an (n-1)-simplex in {displaystyle mathbb {R} ^{n}} with vertices on the co-ordinate axes. Per esempio, suppose n = 3, k = 2 and U is the triangle {displaystyle triangle ABC} in {displaystyle mathbb {R} ^{3}} with vertices A, B and C lying on the {stile di visualizzazione x_{1}} -, {stile di visualizzazione x_{2}} - e {stile di visualizzazione x_{3}} -axes, rispettivamente. The subsets {stile di visualizzazione I} di {stile di visualizzazione {1,2,3}} with exactly 2 elements are {stile di visualizzazione {2,3}} , {stile di visualizzazione {1,3}} e {stile di visualizzazione {1,2}} . Per definizione, {stile di visualizzazione U_{{2,3}}} is the orthogonal projection of {displaystyle U=triangle ABC} onto the {stile di visualizzazione x_{2}X_{3}} -plane, Così {stile di visualizzazione U_{{2,3}}} is the triangle {displaystyle triangle OBC} with vertices O, B and C, where O is the origin of {displaystyle mathbb {R} ^{3}} . Allo stesso modo, {stile di visualizzazione U_{{1,3}}=triangle AOC} e {stile di visualizzazione U_{{1,2}}=triangle ABO} , so the Conant–Beyer theorem says {nome dell'operatore dello stile di visualizzazione {vol} _{2}^{2}(triangle ABC)=nome operatore {vol} _{2}^{2}(triangle OBC)+nome operatore {vol} _{2}^{2}(triangle AOC)+nome operatore {vol} _{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), anche. 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". Il mensile matematico americano. Associazione Matematica d'America. 81 (3): 262–265. doi:10.2307/2319528. JSTOR 2319528. ^ Weisstein, Eric W. "de Gua's theorem". Math World. ^ Howard Whitley Eves: Great Moments in Mathematics (before 1650). Associazione Matematica d'America, 1983, ISBN 9780883853108, S. 37 (excerpt, p. 37, at Google Books) References Weisstein, Eric W. "de Gua's theorem". Math World. 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, Alessandro (2004). "The Theorem of Cosines for Pyramids". Il giornale di matematica del college. Associazione Matematica d'America. 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. Categorie: Theorems in geometryEuclidean geometry