  We introduce a notion of balanced configurations of vectors. This is
motivated by the study of rational A-hypergeometric functions in the sense of
Gelfand, Kapranov and Zelevinsky.
  We classify balanced configurations of seven plane vectors up to GL(2,R)
equivalence and deduce that the only gkz-rational toric four-folds in complex
projective space P^6 are those varieties associated with an essential Cayley
configuration. In this case, we study a suitable hyperplane arrangement and
show that all rational A-hypergeometric functions may be described in terms of
toric residues.
