Balanced Line for a 3-Colored Point Set in the Plane

In this note we prove the following theorem. For any three sets of points in the plane, each of n ≥ 2 points such that any three points (from the union of three sets) are not collinear and the convex hull of 3n points is monochromatic, there exists an integer k ∈ {1, 2, . . . , n− 1} and an open half-plane containing exactly k points from each set.