Putnam 1969 A3

Let $P$ be a non-selfintersecting closed polygon with $n$ sides. Let its vertices be $P_1 , P_2 ,\ldots, P_n .$ Let $m$ other points, $Q_1 , Q_2 ,\ldots, Q_m$ , interior to $P$ , be given. Let the figure be triangulated. This means that certain pairs of the $(n+m)$ points $P_1 ,\ldots , Q_m$ are connected by line segments such that (i) the resulting figure consists exclusively of a set $T$ of triangles, (ii) if two different triangles in $T$ have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in $T$ is precisely the set of the $(n+m)$ points $P_1 ,\ldots , Q_m.$ How many triangles are in $T$ ?