How can we divide a plane with $2n$ points into two equal halves with $n$ points each using a line?
The usual construction is this: since there are only finitely many points, the collection of directions from one point to another is finite. Take any line with a slope which differs from all of those directions. Then no parallel translate of this line can go through $2$ of your points. Start to parallel translate your line toward the side that has more points. As you translate the count changes one at a time so eventually you reach parity.