Why is the Pythagoras theorem true anyway?

[has been a draft for several years, just publishing it because I don't plan on completing it]

I have a list of things I don't really get in math: 

1. the Pythagoras theorem 
2. generating functions // why Fourier transforms are special among transformations // why the normal distribution is the fixed point of the Fourier transform
3. the second fundamental theorem of Lie theory aka the Baker-Campbell-Hausdorff formula
4. Fisher information
5. why the first two derivatives are all that matter in physics // why position and momentum determine the state

(It's mildly interesting that all these things seem to have a unifying theme of "why is the second-order so much more important than anything higher-order?" Perhaps there is a connection between them -- there is probably a connection between 1 and 2, and I think I've read paper about "deriving physical laws from Fisher information" or something like that so maybe there's a connection between 4 and 5.)

The Pythagoras theorem sticks out in this list for being quite elementary, pedagogically. In fact from a historical perspective, it is the most elementary mathematical result in math: in both ancient mathematical traditions, the Indian and the Greek, the Pythagorean theorem was the first mathematical rule to be discovered -- predating (perhaps causing) all other math and science. Yet all its proofs seem like hacks. No, I will not make four copies of that triangle and put them in a square.

One way to think about the Pythagoras theorem is like this: if you line up two segments parallelly and outward, the area of the square of their "hypotenuse" is $(x+y)^2=x^2+y^2+2xy$. If you rotate one of the segments 180 degrees into the same direction as the other, the area of their "hypotenuse" is $x^2+y^2-2xy$. But if you line them up exactly midway between those two extremes, i.e. perpendicularly, then that area is also mid-way between them, i.e. $x^2+y^2$.

Of course, this is not good enough motivation either: there is no reason we should be thinking of areas in the first place, there is no reason why the length couldn't simply be the average of $x+y$ and $x-y$ (I mean OK, it can't because that's just $x$, but you get the point) or something either. But we kind of have sort of a concrete question to think about now: why does this function

The reason why squares come into picture, I think, has to do with the equivariance rules underlying geometry -- specifically, scale invariance.

(It makes sense to reason from invariance rules, because fundamentally, geometry is about the behaviour of a space under transformations. Euclidean geometry, for example, is defined by a plane invariant under translations, rotations and reflections and equivariant under scalings -- trigonometry is invariant under scalings too.)

Namely, any length $h$ constructed from $x$ and $y$ must satisfy:

$$h(\alpha x, \alpha y), = \alpha h(x, y)$$

 So, you can imagine,