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 is the latest proof (demonstration, if we have to be precise) of mine, based on the fact that five points in general position always form a pentagon. There are only two cases to consider.
Case 1. If the pentagon is convex we are done (as four of its vertices form a convex quadrilateral).
Case 2. If the pentagon is concave let us rotate it until one of its edges becomes parallel to the X axis, so that we use the pentagon as a wine glass and pour wine through the diagonal that is outside the pentagon, if this is possible. There exist only four types of glass, which we call Flat Bottom, Flat Top, Flat Middle and No Wine, and for every one of them we can easily construct a convex quadrilateral (see below).