In 1893 J. J. Sylvester [8] posed the following celebrated problem: Given a finite collection of points in the affine plane, not all lying on a line, show that there exists a line which passes through precisely two of the points. Sylvester’s problem was reposed in this Monthly by Erd˝os in 1944 [4] and then later that year a proof was given by Gallai [6]. Since then, many proofs of the Sylvester-Gallai Theorem have been found. Of these proofs, that given by Kelly (as communicated by Coxeter in [2] and [3]) and that attributed to Melchior (as implied in [7]1) are particularly elegant. Kelly’s proof uses a simple distance argument while Melchior considers the dual collection of lines and applies Euler’s formula. For more extensive treatments of the Sylvester-Gallai Theorem and its relatives, see [1] and [5].

Given a collection of points, a line passing through just two of the points is commonly referred to as an ordinary line. As in Melchior [7], one can use projective duality to obtain a fully equivalent dual formulation of the theorem, namely that given a collection of n lines in the real projective plane, not all passing through a common point, there must be a point of intersection of just two lines. Such a point of intersection is generally referred to as an ordinary point. In what follows I provide a new and particularly simple non-metric proof of the theorem in this dual form.

By:* Jonathan Lenchner*

Published in: RC24603 in 2008

**LIMITED DISTRIBUTION NOTICE:**

**This Research Report is available. This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). I have read and understand this notice and am a member of the scientific community outside or inside of IBM seeking a single copy only.**

Questions about this service can be mailed to reports@us.ibm.com .