There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for wich spaces we can do something like?
Edit 1: There are many metric spaces that don't accept some theorems like Pythagorean because we don't have any think like right triangle at all. In my view we need some think like paralled lines and right angel maybe a inner product or else!
Edit 2: Pythagorean like: A releation between $d(a,b)$, $d(a,c)$ and $d(c,b)$. For example: In general for any metric space we have: $d(a,b) \leq d(a,c) + d(c,b)$