  We consider ternary disc packings of the plane, i.e. the packings using discs
of three different radii. Packings in which each ''hole'' is bounded by three
pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$,
$1{>}r{>}s$, allowing triangulated packings by discs of radii 1, $r$ and $s$.
In this paper, we enhance existing methods of dealing with maximal-density
packings in order to find ternary triangulated packings which maximize the
density among all the packings with the same disc radii. We showed for 16 pairs
that the density is maximized by a triangulated ternary packing; for 15 other
pairs, we proved the density to be maximized by a triangulated packing using
only two sizes of discs; for 40 pairs, we found non-triangulated packings
strictly denser than any triangulated one; finally, we classified the remaining
cases where our methods are not applicable.
