A New Proof of the Sylvester-Gallai Theorem

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.