# 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?