Just for Fun: Proving the formula for diagonals of an N-gon

Proof: The number of diagonals of a polygon with N vertices (N-gon) can be expressed by Diagonals=[N(N-3)]/2.

Instead of looking at base cases and taking an inductive approach, let's just pick a vertice.

1) In a polygon with N vertices, we can pick one of any N vertices.
2) From any vertice, diagonals can be drawn to all of the vertices but 3 (or N-3 vertices).
3) Multiplying N(N-3) covers the total number of diagonals from our randomly selected vertice, as well as if we had picked any of the other N-1 vertices to start from.
4) N(N-3), however, is not correct as that counts the diagonal AB and the diagonal BA as different diagonals.
5) Given this overlap, we simply divide N(N-3) by 2.
6) Total # of Diagonals=D=[N(N-3)]/2. QED.

Hope you enjoyed this as much as I did!