On circumscribing polygons for line segments

Abstract Any family of k3 + 1 pairwise disjoint line segments in the Euclidean plane E2, such that no three of their endpoints are collinear, has k + 1 members admitting a circumscribing polygon. That is, one can find a simple polygon P with 2k + 2 vertices such that each of these segments is either an edge or an internal diagonal of P.