  Three hyperbolic type metrics including the triangular ratio metric, the
$j^*$-metric and the M\"obius metric are studied in an annular ring. The
Euclidean midpoint rotation is introduced as a method to create upper and lower
bounds for these metrics, and their sharp inequalities are found. A new
M\"obius-invariant lower bound is proved for the conformal capacity of a
general ring domain by using a symmetric quantity defined with the M\"obius
metric.
