# There are $n$ points in the plane, not all collinear. Prove that there exists a line passing through exactly $2$ points.

I try the method of induction on it, but I fail at the last step. I assume that the statement is true for all planes with less than n points. Then if I add one more point to the plane so that it is not collinear to the line with exaclty two points on it, the statement is true for the plane with n points. However, if the new point is added on the line with exactly two points, how can we make sure that there is still a line passing through exactly two points?

This is called the Sylvester-Gallai Theorem. You can find many proofs on the internet, including in the Wikipedia article:

https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem

See the following notes for a nice and slick proof: http://web.stanford.edu/~yuvalwig/math/teaching/WhatsThePoint.pdf

• In Kelly's proof, what does ordinary of a line mean? – vegetandy Aug 26 at 12:40
• The terminology is "ordinary line" (rather than "ordinary of a line"). This is defined at the beginning of the article; it's a line that goes through just two points in the collection. – halrankard Aug 26 at 12:43
• Thanks, I understand it now, the proof is so clever. – vegetandy Aug 26 at 12:45