I am working through an interesting combinatorial geometry problem book (in my free time, now that it is the summer holiday), and found the following problem which seems easy, but the I can't see the proof of it. It says like this:
Consider $A,B$ two sets of points in plane, with $|A|=2n,|B|=2m,\ m,n \geq 1$ ($|X|$ is the cardinal of the set $X$), such that no three points from $A\cup B$ are collinear. Prove that there is a line which splits the plane in half such that each half contains $n$ points from $A$ and $m$ points from $B$.
When there is only one set, the problem is quite simple. I choose a line which is not parallel with neither of the line formed by the points, put it such that all points are on one side of it and "slide" it across the plane knowing, that only one point at a time can be on the given line, until half of the points are on either side.
This argument doesn't seem to work in this case. Thank you for your help.