Given a set of 9 points in the plane, no three collinear, show that for each point P in the set, the number of triangles containing P formed from the other 8 points in the set must be even.
Solution
Join each pair of points, thus dividing the plane into polygonal regions. If a point P moves around within one of the regions then the number of triangles it belongs to does not change. But if it crosses one of the lines then it leaves some triangles and enters others. Suppose the line is part of the segment joining the points Q and R of the set. Then it can only enter or leave a triangle QRX for some X in the set. Suppose x points in the set lie on the same side of the line QR as P. Then there are 6 - x points on the other side of the line QR. So P leaves x triangles and enters 6-x. Thus the net change is even. Thus if we move P until it is in the outer infinite region (outside the convex hull of the other 8 points), then we change the number of triangles by an even number. But in the outside region it belongs to no triangles.
Note that the same argument works for any odd number of points.
© John Scholes
jscholes@kalva.demon.co.uk
19 Oct 2002