Is d(0,a) = d(b,a+b) true in general?

Question

As an exercise, I was trying to prove, given a distance metric $d$ on a metric space $X$, that if $a,b\in X$ then $d(0, a) = d(b, a+b)$ but I'm not seeing a way to do it. Is this necessarily true in a metric space? No obvious counter-example comes to mind, since it seems to be true in the Euclidean distance metric and in the Manhattan distance metric, and in the distance metric which selects the maximum absolute difference of each coordinate (haven't proved to myself that those are true, but it seems likely at first glance). In case it's helpful, here's some of what I've been toying with:

$d(0, a) \leq d(0, b) + d(b, a)$

$d(a, a+b) \leq d(a,b)+d(b, a+b)$

$d(b, a+b) \leq d(b, a) + d(a, a+b)$

Answer 1

This is true in a normed vector space. $$d(a,b) = \|a - b \| = \|(a+c) - (b+c)\| = d(a+c, b+c) $$

Ok?

Answer 2

In general it is not true. If you define metric on $\mathbb{R} $ by formula $$d(u,v)=|\arctan u -\arctan v|$$ then it is not true.

Answer 3

I realize that the question is ill-formed. I was assuming a structure on the points that isn't necessary in a topology.