The number of generalized balanced lines

Let S be a set of r red points and b = r + 2δ blue points in general position in the plane, with δ ≥ 0. A line l determined by them is balanced if in each open half-plane bounded by l the difference between the number of blue points and red 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.