On the density of iterated line segment intersections

Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: With Si already determined, let Li be the set of all the line segments connecting pairs of points of ⋃i j=1 Sj , and let Si+1 be the set of intersection points of those line segments in Li, which cross but do not overlap. We show that with the exception of some starting configurations the set of all crossing points ⋃ ∞ i=1 Si is dense in a particular subset of the plane with nonempty interior. This region is the intersection of all closed half planes which contain all but at most one point from S1.