Date: Apr 27, 2001 3:47 AM
Subject: Re: Lines in the plane
"S.K.Mody" <firstname.lastname@example.org> wrote in message
> "Mike Schubert" <email@example.com> wrote in message
> > How it can be proved that for n>=3, n distinct points in the plane,
> > not all on a single line, determine at least n distinct lines ?
> Suppose that exactly k of the points lie on the same line
> for some k ( 2 <= k < n ). Then each of the remaining
> n - k points can be paired with each of these points leading
> to (n - k)*k distinct lines. Along with the first line this leads
> to 1 + (n-k)*k lines which is >= n for n >= 3.
This isn't correct. I guess you need to use induction. Assume
that the statement is true for some n ( >= 3 ). Then given
n + 1 points choose n of them which are not all on a single
line. There must be at least n lines formed by these. Of the
lines formed by connecting the (n+1)-th point to each of the
others at least one must be distinct from the original n (the
worst case being when the (n+1)-th point is colinear with
n-1 of the original n). So there are n+1 lines for n+1 points.