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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, 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
How to Cite This Entry:
Pythagorean theorem, multi-dimensional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pythagorean_theorem,_multi-dimensional&oldid=32914
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article