Given 5 coplanar points with no three collinear, prove that there must be a subset of 4 points that form a convex quadrilateral.

Let S be the set of five points.

Let N be the number of points of
S in the boundary of the convex 
hull of S.

With the restriction that no three
are collinear, N must be 3, 4, or 5.

If N=5, then pick any four for the
convex quadrilateral.

If N=4, then those four are the
convex quadrilateral.

If N=3, then we have a triangle with
two of the points inside the triangle.
Those two points determine a line.
With the restriction that no three
are collinear, that line intersects
two sides of the triangle ( does not
pass through a vertex of the triangle ).
Eliminate the vertex which is the 
endpoint of the two sides. The 
remaining four points of S are the
convex quadrilateral.

Posted by Bractals on 2009-10-05 16:25:50