TITLE: which axiom(s) are behind the Pythagorean Theorem
QUESTION [10 upvotes]: There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the (Euclidean geometry) axioms. Are all these proofs equivalent? Do they all track back to the same axioms?

REPLY [2 votes]: If you skipped 10th grade plane geometry because you just loved algebra, your path would take you right to the complex numbers. You don't know any geometry but you like the idea of representing complex numbers 
z = x + yi
as points/vectors in a two coordinate Real Number 'Plane', Of course you naturally put the pure imaginary numbers (y axis) 'orthogonal' to the pure real numbers (x axis).
As you play with your new toy, you begin to recognize geometric type things going on. You decide that no matter what, the 'vectors'
1 + 0i, 0 + 1i, -1 + 0i, 0 - 1i
must all be at a distance of 1 from 0, and you want to create an absolute value (length) for any complex number. You also want the length to be an extension of the absolute value function defined for (pure) real numbers.
If $z = 5i$ and $w = 3$, $zw = 15i$, so you think that, tentatively, you want to keep
$\lvert z w\rvert$ = $\lvert z\rvert \lvert w\rvert$
One day while fooling around you start to examine the conjugates of complex numbers. It then hits you! The distance traveled by going x steps horizontally and then y steps vertically is invariant under the sign of y.
So your 'killer' axiom is:
The absolute value of a complex number is equal to the absolute value of its conjugate:
$\lvert\bar z\rvert = \lvert z\rvert$
But now that is all you need, since
$z \bar z = x^2 + y^2 = \lvert z \bar z\rvert = \lvert z\rvert \lvert\bar z\rvert = \lvert z\rvert ^2$
You define the absolute value of z to be the square root $x^2 + y^2$, and wonder where that will take you.
Ans: The Euclidean 2-Dimensional Plane.
Exercise: Show that with this definition, the absolute value of a product is the product of the absolute values.