We prove a geometric property of the set A^{-1} of inverses of the nonzero elements of an F_q-subspace A of a finite field involving the size of its intersection with two-dimensional F_q-subspaces. We give some applications, including a new upper bound on the size of the intersection of A^{-1} with B when A and B are F_q-subspaces of different dimension of a finite field, satisfying a suitable assumption