How Does Modern Mathematics Prove the Pythagorean Theorem?

[Someone2841]

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:

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

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

 

[micromass]

A proof is always relative to an axiom system and to the definitions you accept. In your proof above you have accepted the definition that length is $d(x,y) = \sqrt{\sum_{k} |x_k-y_k|^2}$. This makes the Pythagorean theorem trivial.

However, if you don't accept that definition of length, but rather some other definition, then the Pythagorean theorem is much more difficult to prove. The usual proofs of the Pythagorean theorem depend on axioms and definitions which don't make the theorem trivial.