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.
Its seems to me that a recursive argument should work.
I will use the word "lines" to mean "lines through exactly two points"
It clearly works for three points  there are three lines.
For four points there are two possibilities:
 no points collinear  6 lines
 one set of three collinear  3 lines
For five points there seem to be four possibilities:
 no points collinear  10 lines
 one set of three collinear  7 lines
 two sets of three collinear  4 lines
 one set of 4 collinear  4 lines
It appears at first that we are guaranteed at least n1 lines for n points and this minimum occurs when all but one of them are collinear.
But then I found a counterexample with 7 points and only 3 lines:
3 points form a equilateral triangle, 3 more at the midpoints, and 1 at the center.

Posted by Jer
on 20110327 23:56:40 