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