A basic property of any linear programming problem with a bounded feasible region is that every feasible solution can be expressed as a convex combination of the CPF solutions (perhaps in more than one way). Similarly, for the augmented form of the problem, every feasible solution can be expressed as a convex combination of the BF solutions.
(a) Show that any convex combination of any set of feasible solutions must be a feasible solution (so that any convex combination of CPF solutions must be feasible).
(b) Use the result quoted in part (a) to show that any convex combination of BF solutions must be a feasible solution.