Sylvester asked the following question: For n points in a plane, not all collinear, is there a line which passes through *exactly* two points?.(Such a line is called an ordinary line or a Gallai line.) The question was asked again by Erdos and answered by Gallai.
The answer is in positive and there are many proofs for the same. Here I outline a beautiful proof given by Kelly.
In a point set P,consider all the connecting lines. Now see the perpendicular distance between a point and a line for all point-line pairs. Let the point p and the line l has the smallest such distance and let q be the point where the perpendicular line from p meets l. We claim that l is a Gallai line.
To prove, assume that l has more than two points. So q has at least two points on one side of it on l. Let p1 and p2 be those points and assume that p2 lies between p1 and q. Now it is easy to see that the line connecting p and p1 and the point p2 has a smaller distance than the p-l pair, which is a contradiction. Hence, proved.
