The number of generalized balanced lines

Abstract Let S be a set of r red points and b = r + 2δ blue points in general position in the plane. A line determined by them is said to be balanced if in each open half-plane bounded by the difference between the number of red points and blue points is δ. We show that every set S as above has at least r balanced lines. The main techniques in the proof are rotations and a generalization, sliding rotations, introduced here