  Let $G \subsetneq \mathbb{R}^n$ be a domain and let $d_1$ and $d_2$ be two
metrics on $G$. We compare the geometries defined by the two metrics to each
other for several pairs of metrics. The metrics we study include the distance
ratio metric, the triangular ratio metric and the visual angle metric. Finally
we apply our results to study Lipschitz maps with respect to these metrics.
