*Given any five points on a flat surface that are in general position, i.e. no two of them coinciding and no three of them in a straight line, prove that four of these points will always form a convex quadrilateral.*

Let us start with the observation that a line can intersect three sides of a triangle at most (see below).

*T1*and

*T2*, there always exist a pair of sides

*s1*

*∊T1*and

*s2*

*∊T2*, such that neither

*s1*nor its extension intersects

*s2*(and vice versa). Let us call such a pair ‘useful’ (see the black pair on the picture below).