If there are an even number of points in a plane, assuming no three are collinear, how many ways can they be divided into equal halves by a line? This would be counting the number of possible partitions of the points, and not the (infinite) number of lines.
Further, on what conditions does collinearity of points reduce this quantity? In the extreme case, when all points are along a line, there's only one way to split them: down the middle. Is there some inductive method that can be used to determine this quantity given all the points?