linear programming proof

May 2010
Let A belong to R^m×n, b belong to Rm, and let c belong to R^n. Show that,
if max(cTx : Ax = b, x >= 0) has two solutions, then it has infinitely many solutions.

I'm having trouble formally proving this problem.
I understand that if there is two maximum points then there must be an infinite number of x vectors but how can I prove that this results in an infinite number of maximum points?