I think it would be fun to 'prove' the Pythagorean theorem in the following way, using only a small number of axioms that relate the physical world to the world of mathematics.
Assume that the collection of lengthts in the physical world corresponds one-to-one to the set of real numbers $\mathbb R$, and that the (physical) Euclidean plane therefore should be thought of as the vector space $\mathbb R^2$, as each point in the Euclidean plane corresponds uniquely to a pair of lengths (namely vertical and horizontal). These statements can be motivated physically and we'll take them as axioms.
Now my idea is that there might also be a physical motivation for requiring that the norm on $\mathbb R^2$ (modeling the physical Euclidean plane) come from some inner product. And if we take that as an axiom, then there is an elementary result about Hilbert spaces that says that $||x+y|| = ||x||+||y||$ whenever $x,y\in\mathbb R^2$ are orthonormal, which we may interpret as the Pythagorean theorem.
So if anyone knows a physical reason that the norm (representing lengths) should come from an inner product, I'd love to hear it.
N.B. I wasn't sure if I should post this question here or in the physics community, but I think mathematicians will have thought more about the axioms of inner products and things like that, so I think this is the right place.