Pythagorean theorem, multi-dimensional

Consider the $n$-dimensional space $\mathbf R^n$ (with the usual metric and measure). Let $A_i$ be a point on the $i$th coordinate axis and let $O$ be the origin. Let $s$ be the $(n-1)$-dimensional volume of the $(n-1)$-dimensional simplex $A_1\ldots A_n$ and let $s_i$ be the $(n-1)$-dimensional volume of the $(n-1)$-dimensional simplex $OA_1\ldots A_{i-1}A_{i+1}\ldots A_n$. Then $s^2=\sum_{i=1}^ns_i^2$.

For other and further generalizations of the classical Pythagoras theorem, see [a2] and the references therein.

References

[a1]	Etsua Yoshinaga, Shigeo Akiba, "Very simple proofs of the generalized Pythagorean theorem" Sci. Reports Yokohama National Univ. Sect. I , 42 (1995) pp. 45–46
[a2]	D.R. Conant, W.A. Beyer, "Generalized Pythagorean theorem" Amer. Math. Monthly , 81 (1974) pp. 262–265