de Gua's theorem

The Pythagorean theorem and de Gua's theorem are special cases ({{math|1=n = 2, 3}}) of a general theorem about n-simplices with a right-angle corner, proved by P. S. Donchian and H. S. M. Coxeter in 1935.{{cite journal |last1=Donchian |first1=P. S. |last2=Coxeter |first2=H. S. M. |date=July 1935 |title=1142. An n-dimensional extension of Pythagoras' Theorem |journal=The Mathematical Gazette |volume=19 |issue=234 |pages=206 |doi=10.2307/3605876|jstor=3605876 |s2cid=125391795 }} This, in turn, is a special case of a yet more general theorem by Donald R. Conant and William A. Beyer (1974),{{cite journal |doi=10.2307/2319528 |title=Generalized Pythagorean Theorem |author1=Donald R Conant |author2=William A Beyer |name-list-style=amp |journal=The American Mathematical Monthly |volume=81 |date=Mar 1974 |pages=262–265 |jstor=2319528 |issue=3 |publisher=Mathematical Association of America }} which can be stated as follows.

Let U be a measurable subset of a k-dimensional affine subspace of $\backslash mathbb\{R\}^n$ (so $k\; \backslash le\; n$). For any subset $I\; \backslash subseteq\; \backslash \{\; 1,\; \backslash ldots,\; n\; \backslash \}$ with exactly k elements, let $U\_I$ be the orthogonal projection of U onto the linear span of $e\_\{i\_1\},\; \backslash ldots,\; e\_\{i\_k\}$, where $I\; =\; \backslash \{i\_1,\; \backslash ldots,\; i\_k\backslash \}$ and $e\_1,\; \backslash ldots,\; e\_n$ is the standard basis for $\backslash mathbb\{R\}^n$. Then

$$\backslash operatorname\{vol\}\_k^2(U)\; =\; \backslash sum\_I\; \backslash operatorname\{vol\}\_k^2(U\_I),$$

where $\backslash operatorname\{vol\}\_k(U)$ is the k-dimensional volume of U and the sum is over all subsets $I\; \backslash subseteq\; \backslash \{\; 1,\; \backslash ldots,\; n\; \backslash \}$ 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 $\backslash mathbb\{R\}^n$ with vertices on the co-ordinate axes. For example, suppose {{math|1=n = 3}}, {{math|1=k = 2}} and U is the triangle $\backslash triangle\; ABC$ in $\backslash mathbb\{R\}^3$ with vertices A, B and C lying on the $x\_1$-, $x\_2$- and $x\_3$-axes, respectively. The subsets $I$ of $\backslash \{\; 1,\; 2,\; 3\; \backslash \}$ with exactly 2 elements are $\backslash \{\; 2,3\; \backslash \}$, $\backslash \{\; 1,3\; \backslash \}$ and $\backslash \{\; 1,2\; \backslash \}$. By definition, $U\_\{\backslash \{\; 2,3\; \backslash \}\}$ is the orthogonal projection of $U\; =\; \backslash triangle\; ABC$ onto the $x\_2\; x\_3$-plane, so $U\_\{\backslash \{\; 2,3\; \backslash \}\}$ is the triangle $\backslash triangle\; OBC$ with vertices O, B and C, where O is the origin of $\backslash mathbb\{R\}^3$. Similarly, $U\_\{\backslash \{\; 1,3\; \backslash \}\}\; =\; \backslash triangle\; AOC$ and $U\_\{\backslash \{\; 1,2\; \backslash \}\}\; =\; \backslash triangle\; ABO$, so the Conant–Beyer theorem says

$$\backslash operatorname\{vol\}\_2^2(\backslash triangle\; ABC)\; =\; \backslash operatorname\{vol\}\_2^2(\backslash triangle\; OBC)\; +\; \backslash operatorname\{vol\}\_2^2(\backslash triangle\; AOC)\; +\; \backslash operatorname\{vol\}\_2^2(\backslash triangle\; ABO),$$

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.

De Gua's theorem can also be generalized to arbitrary tetrahedra and to pyramids, similarly to how the law of cosines generalises Pythagoras' theorem.{{cite journal |last1=Kheyfits |first1=Alexander |year=2004 |title=The Theorem of Cosines for Pyramids |journal=The College Mathematics Journal |publisher=Mathematical Association of America |volume=35 |issue=5 |pages=385–388 |doi=10.2307/4146849 |jstor=4146849}}{{Cite journal |last=Tran |first=Quang Hung |date=2023-08-02 |title=A Generalization of de Gua's Theorem with a Vector Proof |url=https://link.springer.com/10.1007/s00283-023-10288-0 |journal=The Mathematical Intelligencer |language=en |doi=10.1007/s00283-023-10288-0 |issn=0343-6993}}

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).{{MathWorld|title=de Gua's theorem|urlname=deGuasTheorem}}Howard Whitley Eves: Great Moments in Mathematics (before 1650). Mathematical Association of America, 1983, {{ISBN|9780883853108}}, S. 37 ({{Google books|9_w5jDPTvCQC|excerpt|page=37}})

• Vector area and projected area
• Bivector

• Sergio A. Alvarez: [http://www.cs.bc.edu/~alvarez/NDPyt.pdf Note on an n-dimensional Pythagorean theorem], Carnegie Mellon University.
• {{cite journal |last1=Hull |first1=Lewis |last2=Perfect |first2=Hazel |last3=Heading |first3=J. |year=1978 |title=62.23 Pythagoras in Higher Dimensions: Three Approaches |journal=Mathematical Gazette |volume=62 |number=421 |pages=206–211 |jstor=3616695 |doi=10.2307/3616695 |s2cid=187356402 }}
• {{MathWorld|title=de Gua's theorem|urlname=deGuasTheorem}}

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Category:Theorems about polyhedron

Category:Theorems in geometry

Category:Euclidean geometry