De Gua's theorem [SEP] named after
De Gua's theorem
De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named after Jean Paul de Gua de Malves.
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.
Generalizations.
The Pythagorean theorem and de Gua's theorem are special cases ("n" =
), as well. However the theorem had also been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650).
References.
- 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.
- Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.