Putnam 2007 A6

A triangulation $\mathcal{T}$ of a polygon $P$ is a finite collection of triangles whose union is $P$, and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side is a side of exactly one triangle in $\mathcal{T}$. Say that $\mathcal{T}$ is admissible if every internal vertex is shared by 6 or more triangles. For example, [figure omitted.] Prove that there is an integer $M_n$, depending only on $n$, such that any admissible triangulation of a polygon $P$ with $n$ sides has at most $M_n$ triangles.