De Gua's theorem

De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named for 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.

$A_{\text{yellow}}^{2}=A_{\text{green}}^{2}+A_{\text{blue}}^{2}+A_{\text{red}}^{2}\,$

front - face opposite of the right angle corner	back - faces at right angle corner	view from the side

The Pythagorean theorem and de Gua's theorem are special cases (n = 2, 3) of a general theorem about n-simplexes with a right angle corner.

References

• Weisstein, Eric W. "de Gua's theorem". MathWorld.
• Note on a n-dimensional Pythagorean theorem (PDF)
• De Gua's theorem and other analog theorems