KURSCHAK 1985 Problem 1

We have triangulated a convex $(n+1)$ -gon $P_0P_1\dots P_n$ (i.e., divided it into $n-1$ triangles with $n-2$ non-intersecting diagonals). Prove that the resulting triangles can be labelled with the numbers $1,2,\dots,n-1$ such that for any $i\in\{1,2,\dots,n-1\}$ , $P_i$ is a vertex of the triangle with label $i$ .