This solution is by John E. Ferguson of Oregon State University:
Any straight line must contain at least four points P1, P2, P3, P4 that fall into one of the sets (this must be true of one set or the other). Call that set S1. Now consider the configuration above (the distances between the points are out of scale). If S1 doesn’t contain the vertices of a rectangle, then at least three points in the A group and at least three points in the B group must belong to the other set, S2. And in that case there is an S2 rectangle.