Pythagoras' theorem

Pythagoras' theorem is a natural consequence of having the typical metric on a plane. However, in his proof, he never mentions what metric he uses - somehow, he must have snuck the metric in. Where does his proof assume the usual metric?

http://en.wikipedia.org/wiki/Pythagorus's_theorem#Proof_using_similar_triangles

>>8
Notice how I mentioned "pseudoriemannian manifolds" instead of "riemannian manifolds".  Inner product spaces on the former are not positive definite.

http://en.wikipedia.org/wiki/Triangle_inequality (near the bottom) shows that the triangle inequality reverses (with some constraints) in the Minkowski metric (or "metric" if you prefer).  It also includes Lorentz invariance.