```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 messagenews://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. Assumethat the statement is true for some n ( >= 3 ). Then givenn + 1 points choose n of them which are not all on a singleline. There must be at least n lines formed by these. Of thelines formed by connecting the (n+1)-th point to each of theothers at least one must be distinct from the original n (theworst case being when the (n+1)-th point is colinear withn-1 of the original n). So there are n+1 lines for n+1 points.Regards,S.K.Mody.
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