At least in Euclidean geometry and the upper half plane model of hyperbolic geometry, the statements '$y$ lies on the line segment determined by $x$ and $z$ ' and '$d(x,y)+d(y,z)=d(x,z) $' are equivalent.
I wonder whether this characterization holds in other geometries and whether it has a name to it. Do geometries with this relation have some special properties?
I also think it is somehow related to the notion of uniform convexity in Banach spaces, but that is a different problem since we don't ask for vector space sturcture here.
Any insight or reference would be helpful! Thanks!