View source for Pythagorean theorem, multi-dimensional

Consider the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102801.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102802.png" /> (with the usual metric and measure). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102803.png" /> be a point on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102804.png" />th coordinate axis and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102805.png" /> be the origin. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102806.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102807.png" />-dimensional volume of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102808.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102809.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028010.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028011.png" />-dimensional volume of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028012.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028013.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028014.png" />.

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>