In this paper three sufficient conditions are provided under each of which an upper semicontinuous point-to-set mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope. Furthermore, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary polytopes. These results follow in a constructive way by designing a new simplicial algorithm. The algorithm operates on a triangulation of the polytope and generates a piecewise linear path of points connecting two distinct faces of the polytope. Each point on the path is an approximate zero point. As the mesh size of the triangulation goes to zero, the path converges to a connected set of zero points linking the two distinct faces. As a consequence, our results generalize Browder's fixed point theorem [ Summa Brasiliensis Mathematicae, 4 (1960), pp. 183-191] and an earlier result by the authors [ Math. Oper. Res., 21 (1996), pp. 675-696] on the n-dimensional unit cube. An application in economics and some numerical examples are also discussed.