  A new intrinsic metric called $t$-metric is introduced. Several sharp
inequalities between this metric and the most common hyperbolic type metrics
are proven for various domains $G\subsetneq\mathbb{R}^n$. The behaviour of the
new metric is also studied under a few examples of conformal and quasiconformal
mappings, and the differences between the balls drawn with all the metrics
considered are compared by both graphical and analytical means.
