Triangle inequality in the Poincaré half plane

Question

How can I prove the triangle inequality holds in the Poincaré Half Plane when given points $A$, $B$, $C$. Using the idea:

$$d(A,B) + d(B,C) \ge d(A,C),$$

where $d(A,B)$ is the Poincaré distance from $A$ to $B$.