View source for Pythagorean theorem, multi-dimensional

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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|Pythagoras theorem]], see [[#References|[a2]]] and the references therein.

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