If a finite set of n>2 points in the plane are not all on one line, then prove that there exists a line through exactly two of the points.

If one looks at this combinatorially, there are n(n-1)/2 choices for pairs of points, and each pair is on exactly one line. Thus, it is enough to prove that there must be more than n(n-1)/6 lines, as then some line has fewer than 3 pairs (and thus must only have 1 pair)

My guess is this would be guaranteed by the geometry somehow, but I'm not sure myself.

It seems like some sort of inductive argument might give this as well.

Posted by Gamer on 2011-03-28 02:48:08