Lesson 2

The proof by induction proceeds as follows. The base case \(n=3\) is easily verified. For \(n>3\), consider an arbitrary diagonal of the triangulation. It divides the \((n+1)\)-gon into a convex \(k\)-gon and a convex \(l\)-gon, where \(k+l=n+3\) (the two vertices of this diagonal are vertices of both the \(k\)-gon and the \(l\)-gon). By the induction hypothesis, the number of triangles is \[(k-2)+(l-2)=k+l-4=n+3-4=n-1.\]