  Motivated by problems involving triangle-decompositions of graphs, we examine
the facet structure of the cone $\tau_n$ of weighted graphs on $n$ vertices
generated by triangles. Our results include enumeration of facets for small
$n$, a construction producing facets of $\tau_{n+1}$ from facets of $\tau_n$,
and an arithmetic condition on entries of the normal vectors. We also point out
that a copy of $\tau_n$ essentially appears via the perimeter inequalities at
one vertex of the metric polytope.
