德古阿定理

德古阿定理是勾股定理的三维形式，以法国数学家让·保罗·德古阿·德马尔弗（Jean Paul de Gua de Malves）命名。

De gua theorem 1.svg

若从一个正六面体截下一角O形成一截角锥（记为ABCO），则

S_{ABC}^{2}=S_{\color {blue}ABO}^{2}+S_{\color {green}ACO}^{2}+S_{\color {red}BCO}^{2}

参考资料

埃里克·韦斯坦因. de Gua's theorem. MathWorld.
Sergio A. Alvarez: Note on an n-dimensional Pythagorean theorem （页面存档备份，存于互联网档案馆）, Carnegie Mellon University
De Gua's Theorem, Pythagorean theorem in 3-D （页面存档备份，存于互联网档案馆） - graphical illustration and related properties of the tetrahedron

扩展阅读

Kheyfits, Alexander. The Theorem of Cosines for Pyramids. The College Mathematics Journal (Mathematical Association of America). 2004, 35 (5): 385–388. JSTOR 4146849. Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.