Do these points lie on one line of this metric space?

Question

Let $A, B$, and $C$ be three points in a metric space. Suppose $AB=5, BC=7,$ and $AC=10$ Is it possible that these points lie on one line of this metric space?

Answer 1

I presume $AB$ is shorthand for $d(A, B)$. For the first two commenters, you can give the following definition: three points $a, b, c$ in a metric space are collinear if the triangle inequality is an equality, that is,

$$d(a, b) + d(b, c) = d(a, c)$$

and in this case we say that $b$ is between $a$ and $c$. With this definition the answer to the OP's question is clear. Jelly, if you're using a different definition it would be good to indicate so. You can see these definitions, for example, in this paper.