Property of distance functions given the "homogeneity" axiom

Question

Consider a distance function $D:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ that, in addition to the positive definiteness, symmetry and triangle inequality, satisfies a the following homogeneity axiom: $\forall \lambda>0$ and $\forall x,y\in\mathbb{R}^n$, we have $D(\lambda x,\lambda y)=\lambda D(x,y)$. I have seen a source claiming that, for $\lambda\in[0,1]$ and $x,y\in\mathbb{R}^n$, there is $D(x,y)=D(x,z)+D(z,y)$ given $z=\lambda x+(1-\lambda) y$... while this claim should be false since I am able to construct a counterexample when $n=1$, I am curious where I can find additional studies on this type of distance functions (especially their name if they have one).