Term Paper : “ the Happy Ending Problem ” Due April 12 , 2011

At one of their meetings, Klein challenged the group to solve a complex problem in planar geometry. Klein proposed that the group consider five points on a flat surface, where no three of the five formed a straight line. It was obvious to all that when four of the points were joined, they formed a quadrilateral, but Klein also noticed that given five points, four of them always appeared to define a convex quadrilateral. After the group failed to prove her proposition, Klein offered an informal proof, as follows. She deduced three ways in which a convex polygon could enclose all five points. Her first, and simplest case, was when four points forming a quadrilateral enclosed the fifth point, thereby automatically satisfying the requirement. In the second case, Klein explained that if the convex polygon was a pentagon, any four of five points could be joined to form a quadrilateral and satisfy the requirement. Finally, if three of the points create a triangle, it was obvious that two remained inside the triangle. Klein reasoned that the two points left inside defined a line that split the triangle such that two of the triangle’s points were on one side of the line. It followed that these two exterior points, plus the two interior points automatically formed a convex quadrilateral.[IP00]