Given any five points on a flat surface, with no three of them in a straight line, prove that four of these points will always form a convex quadrilateral.
This is my proof of Esther Klein’s theorem based on the 2+2+1 partition of 5. Let us think about the five points as of two lines and an extra point. Line l1 is defined by P1 and P2, line l2 is defined by P3 and P4 and P5 is the extra point. As any of l1 and l2 divides the plane into two halves, there are more points (not on the line) in the one half of the plane than in the other half (according to the Pigeonhole principle). Let us call the former half of the plain more populated and the latter less populated. There are only 3 cases to consider:
Case 1. Neither line intersects the segment between the points on the other line. One can build a convex quadrilateral by connecting the points on l1 and l2.