De Gua's theorem explained

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: A_^2 = A_^2+A_^2+A_^2 De Gua's theorem can be applied for proving a special case of Heron's formula.[1]

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.[2] This, in turn, is a special case of a yet more general theorem by Donald R. Conant and William A. Beyer (1974),[3] which can be stated as follows.

Let U be a measurable subset of a k-dimensional affine subspace of

Rn

(so

k\len

). For any subset

I\subseteq\{1,\ldots,n\}

with exactly k elements, let

UI

be the orthogonal projection of U onto the linear span of
e
i1

,\ldots,

e
ik
, where

I=\{i1,\ldots,ik\}

and

e1,\ldots,en

is the standard basis for

Rn

. Then\operatorname_k^2(U) = \sum_I \operatorname_k^2(U_I),where

\operatorname{vol}k(U)

is the k-dimensional volume of U and the sum is over all subsets

I\subseteq\{1,\ldots,n\}

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

Rn

with vertices on the co-ordinate axes. For example, suppose, and U is the triangle

\triangleABC

in

R3

with vertices A, B and C lying on the

x1

-,

x2

- and

x3

-axes, respectively. The subsets

I

of

\{1,2,3\}

with exactly 2 elements are

\{2,3\}

,

\{1,3\}

and

\{1,2\}

. By definition,

U\{

} is the orthogonal projection of

U=\triangleABC

onto the

x2x3

-plane, so

U\{

} is the triangle

\triangleOBC

with vertices O, B and C, where O is the origin of

R3

. Similarly,

U\{

} = \triangle AOC and

U\{

} = \triangle ABO, so the Conant–Beyer theorem says

\operatorname_2^2(\triangle ABC) = \operatorname_2^2(\triangle OBC) + \operatorname_2^2(\triangle AOC) + \operatorname_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.

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]

See also

Notes

References

Notes and References

  1. Lévy-Leblond . Jean-Marc . 2020 . The Theorem of Cosines for Pyramids . The Mathematical Intelligencer . SpringerLink. 10.1007/s00283-020-09996-8 . 224956341 . free .
  2. Donchian . P. S. . Coxeter . H. S. M. . July 1935 . 1142. An n-dimensional extension of Pythagoras' Theorem . The Mathematical Gazette . 19 . 234 . 206 . 10.2307/3605876. 3605876 . 125391795 .
  3. 10.2307/2319528 . Generalized Pythagorean Theorem . Donald R Conant . William A Beyer . amp . The American Mathematical Monthly . 81 . Mar 1974 . 262–265 . 2319528 . 3 . Mathematical Association of America .
  4. Kheyfits . Alexander . 2004 . The Theorem of Cosines for Pyramids . The College Mathematics Journal . Mathematical Association of America . 35 . 5 . 385–388 . 10.2307/4146849 . 4146849.
  5. Tran . Quang Hung . 2023-08-02 . A Generalization of de Gua's Theorem with a Vector Proof . The Mathematical Intelligencer . en . 10.1007/s00283-023-10288-0 . 0343-6993.
  6. Howard Whitley Eves: Great Moments in Mathematics (before 1650). Mathematical Association of America, 1983,, S. 37