Editing Simplex (section)

=== Simplices with an "orthogonal corner" ===
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent [[Face (geometry)|faces]] are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an {{mvar|n}}-dimensional version of the [[Pythagorean theorem]]: The sum of the squared {{math|(''n'' − 1)}}-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared {{math|(''n'' − 1)}}-dimensional volume of the facet opposite of the orthogonal corner.
: <math> \sum_{k=1}^n |A_k|^2 = |A_0|^2 </math>
where <math> A_1 \ldots A_n </math> are facets being pairwise orthogonal to each other but not orthogonal to <math>A_0</math>, which is the facet opposite the orthogonal corner.<ref>{{cite journal |last1=Donchian |first1=P. S. |last2=Coxeter |first2=H. S. M. |date=July 1935 |title=1142. An n-dimensional extension of Pythagoras' Theorem |journal=The Mathematical Gazette |volume=19 |issue=234 |pages=206 |doi=10.2307/3605876|jstor=3605876 |s2cid=125391795 }}</ref>

For a 2-simplex, the theorem is the [[Pythagorean theorem]] for triangles with a right angle and for a 3-simplex it is [[de Gua's theorem]] for a tetrahedron with an orthogonal corner.