The number of generalized balanced lines

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