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: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
(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
where
is the
k-dimensional volume of
U and the sum is over all subsets
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
with vertices on the
co-ordinate axes. For example, suppose, 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,
} = \triangle AOC and
} = \triangle ABO, so the Conant–Beyer theorem says
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
- 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 . 3616695 . 10.2307/3616695 . 187356402 .
Notes and References
- Lévy-Leblond . Jean-Marc . 2020 . The Theorem of Cosines for Pyramids . The Mathematical Intelligencer . SpringerLink. 10.1007/s00283-020-09996-8 . 224956341 . free .
- 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 .
- 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 .
- 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.
- 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.
- Howard Whitley Eves: Great Moments in Mathematics (before 1650). Mathematical Association of America, 1983,, S. 37