MATHEMATICAL APPLICATIONS 397 Ax c T · x with x R n , all vectors from R m+1 , and all vectors u R m+1 + 0 m 1 that make an angle with that is not larger than . To prove Farkas's lemma, it remains to write out the condition K = R m+1 , that is, as v = 0 m 1 is an interior point of K, the condition that -v does not belong to K. This gives the condition that the system Ax 0 m , c T · x -1 has a solution. This is seen to be equivalent to the first alternative.2 9.4 THE PROBLEM OF APOLLONIUS