There are lots of geometric dissection and reassembling proofs of the Pythagorean Theorem (PT). I want to know what are the necessary ideas to allow a proof using discrete instances.
For example, we can, not too terribly painstakingly, using elementary tiling of a diagram show that a 1,1 right triangle has area 2 square on the hypotenuse
and even a 3,4 right triangle has 5 as its hypotenuse all without using PT.
Could these instances (or others) be used prove the full generality of PT, assuming some ideas of continuity, some light inference on the relations of the three values?
The idea is simply one of analogy: two polynomials of degree $d$ are equal if they coincide on $d+1$ points. Here, instead of 2 variables ($x$ and $y$) we have 3 ($a,b,c$).
So for a (very loosely analogous) proof of PT, what would be needed?
- what assumption of form of the relation? $d a^2 + e b^2 = f c^2$ seems like too much to assume, practically assuming PT! (but if that's what it takes...)
- how many instances? (obviously 3 for the three vars $d,e,f$, but can you do it in fewer?
- any continuity axioms? (or minor handwaving justifications)