I am wondering what it means to "prove" the Pythagorean Theorem in modern mathematics. Most real analysis begins with the following definition of the n-dimensional Euclidean metric:(adsbygoogle = window.adsbygoogle || []).push({});

##d(x,y) = |y-x| = \sqrt{\sum\limits_{i=0}^{n} (y_i-x_i)^2}##

This would seem to directly imply the Pythagorean theorem!

##d(<0,0>,<a,b>) = |<a,b>| = \sqrt{a^2+b^2} \implies |<a,b>|^2 = a^2+b^2##

What bearing do these have, then: http://en.wikipedia.org/wiki/Pythagorean_theorem#Proofs

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# Proving the Pythagorean Theorem

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