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Re: Lines in the plane
Posted:
Apr 27, 2001 3:47 AM


"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 + (nk)*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 n1 of the original n). So there are n+1 lines for n+1 points.
Regards, S.K.Mody.



