There are five points on a flat surface, no two of which are coinciding and no three of which are on a straight line. Prove that four of these points are vertices of a convex quadrilateral.
This simple theorem has a simple proof but, as the unheard of and complex concept of the so called convex hull is involved, the proof could be unintelligible to non-mathematicians, high school students inclusive. So, here is the simplest proof of mine, in which you can meet the familiar points and lines only.
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. 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.