Why is pythagoras theorem what it is?

[blah]

Why is the absolute distance sqrt(x^2 + y^2)?

I've found several proofs on the internet but none really tell me much.
I believe I have noticed something about pythagoras theorem and that is it allows all inertial reference frames in physics to be equivalent. In physics you can describe objects in any inertial frame and they will have the same relative speeds and distances (in Newtonian physics) and there is no reason to say one inertial frame is more correct than any other. But say the absolute distance was x + y instead of sqrt(x^2 + y^2), then relative speeds and distances would change has you rotated the frame of reference and the laws of physics wouldn't be equivalent in all reference frames. Also, I believe you can say special relativity extends pythagoras therem to include time, and SR finds a way to make time and distances change with velocity and still keep all inertial reference frames equiviilent.

Is this right? and if so why?

 

[dextercioby]

It's correct.Euclidean space is rotationally symmetric and for a vector its modulus (given the Pythagorean theorem in 3 dim-s) is constant both under space translations & under space rotations...

As for flat Minkowski space time,it's the (relativistic) interval that is constant under the rotation group SO(1,3).

Daniel.