I was playing around with Pythagoras's Theorem, and the proofs I was looking at involved areas and congruent triangles. I was wondering how much of this machinery we actually need, and especially if we could do the same thing in a finite space.
For it to even make sense, we'd likely need:
- a (possibly finite) set $P$ of points,
- a set $B$ of line segments (which are subsets of $P$),
- a length function $\ell:B \rightarrow \mathbb{R}$ for line segments,
- the concept of a triangle (which presumably could be defined as sets of three distinct line segments {A,B,C} for which $A \cap B$, $A \cap C$ and $B \cap C$ are unique singletons sets),
- the concept of a "right angle triangle" (which seems to be begging to be defined as any triangle {A,B,C} for which $\ell(A)^2 + \ell(B)^2 = \ell(C)^2$, where I'd be taking the Pythagorean Theorem as axiomatic).
You could define such a "Pythagorean geometry" this way, but I'm not sure if you'd get anything interesting in return for doing so. Hence...
Question: Are there interesting finite geometries with angles or triangles which satisfy a Pythagorean-like identity?
Here, I mean "interesting" in the sense of there's some reason to study them, or perhaps some papers in this general direction. I'm not really sure; I didn't find any from an initial Google or Google Scholar search. I'm mostly just after proof of concept.