On inducing polygons and related problems

Bose et al. (2003) [2] asked whether for every simple arrangement AA of n lines in the plane there exists a simple n-gon P that induces  AA by extending every edge of P   into a line. We prove that such a polygon always exists and can be found in O(nlogn) time. In fact, we show that every finite family of curves CC such that every two curves intersect at least once and finitely many times and no three curves intersect at a single point possesses the following Hamiltonian-type property: the union of the curves in CC contains a simple cycle that visits every curve in CC exactly once.