How to prove triangle inequality for the metric $d(x,y)=\frac{2|x-y|}{\sqrt{\left(1+|x|^2\right)}+\sqrt{\left(1+|y|^2\right)}}$

Question

How to prove triangle inequality for the metric $$d(x,y)=\frac{2|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}},$$ for all $x,y\in \mathbb{C}?$