Consider a k-sided polygon. By assumption it has k(k - 3)/2 diagonals. If we place a triangle on a side AB of the polygon, we make it into a (k + 1)-sided polygon. It has all the diagonals of the k-sided polygon plus the diagonals drawn from the new vertex N to all the
 

vertices of the previous k-sided polygon except 2, namely A and B. In addition, the former side AB has become a diagonal of the new (k + 1)-sided polygon. Thus a (k + 1)-sided polygon has a total ofdiagonals. But:

This is the desired formula for n = k + 1.

So, by assuming that the formula D_n = n(n - 3)/2 is true for n = k, we have been able to show it true for n = k + 1. This, in addition to the fact that it is true for n = 3, proves that it is true for all integers greater than or equal to 3. (The reader may have discovered a more direct method of obtaining the above formula.)
 
 

Saturday, April 11, 1998