Date: Apr 27, 2001 3:47 AM
Author: S.K.Mody
Subject: Re: Lines in the plane

"S.K.Mody" <modysk@hotmail.com> wrote in message

news://tehbrr448ml7bb@news.supernews.com...

> "Mike Schubert" <mikeschub2@iol.com> wrote in message

> news://yifcfwkp2kcn@forum.mathforum.com...

> > 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.

Regards,

S.K.Mody.