  The distortion of six different intrinsic metrics and quasi-metrics under
conformal and quasiregular mappings is studied in a few simple domains
$G\subsetneq\mathbb{R}^n$. The already known inequalities between the
hyperbolic metric and these intrinsic metrics for points $x,y$ in the unit ball
$\mathbb{B}^n$ are improved by limiting the absolute values of the points $x,y$
and the new results are then used to study the conformal distortion of the
intrinsic metrics. For the triangular ratio metric between two points
$x,y\in\mathbb{B}^n$, the conformal distortion is bounded in terms of the
hyperbolic midpoint and the hyperbolic distance of $x,y$. Furthermore,
quasiregular and quasiconformal mappings are studied, and new sharp versions of
the Schwarz lemma are introduced.
