Let $(X,d)$ be a complete metric space with this property:
for each $x \in X$, $r > 0$ and $y \in X$ with $d(x,y) < r$, there exists $z \in X$ such that $d(x,y)+d(y,z) = d(x,z) = r$. I want to know if the family of complete metric spaces with this property are known or some work have been done on it.
Remark: I studied about metrically convex space which had been introduced by K.Menger(1928) and I am not sure if there is a relation between this family with my mentioned metric spaces.