  We prove that any planar 4-web defines a unique projective structure in the
plane in such a way that the leaves of the foliations are geodesics of this
projective structure. We also find conditions for the projective structure
mentioned above to contain an affine symmetric connection, and conditions for a
planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric
surface. Similar results are obtained for planar d-webs, d > 4, provided that
additional d-4 second-order invariants vanish.
