 the sylvesterâgallai theorem asserts that given a finite number of points in the euclidean plane, either # all the points are collinear; or # there is a line which contains exactly two of the points. this claim was posed as a problem by .  suggests that sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve in the complex projective plane form a configuration of nine points and twelve lines in which each line determined by two of the points contains a third point. the sylvesterâgallai theorem implies that it is impossible for all nine of these points to have real coordinates.  claimed to have a short proof, but it was already noted to be incomplete at the time of publication.  proved the projective dual of this theorem, (actually, of a slightly stronger result). unaware of melchior's proof,  again stated the conjecture, which was proved first by tibor gallai, and soon afterwards by other authors.; . a line that contains exactly two of a set of points is known as an ordinary line. there is an algorithm that finds an ordinary line in a set of n points in time proportional to n log n in the worst case.; . == projective and dual versions == the question of the existence of an ordinary line can also be posed for points in the real projective plane rp2 instead of the euclidean plane. this provides no additional generality, as any finite set of projective points can be transformed into a euclidean point set preserving all ordinary lines; but the projective viewpoint allows certain configurations to be described more easily. by projective duality, the existence of an ordinary line in a set of non-collinear points in rp2 is equivalent to the existence of an ordinary point in a nontrivial arrangement of finitely many lines. an arrangement is said to be trivial when all its lines pass through a common point, and nontrivial otherwise; an ordinary point is a point that belongs to exactly two lines. ==proofs== ===kelly's proof=== right for a description of gallai's original proof of the theorem, see e.g. . the proof below is instead due to kelly. suppose for contradi
Euclidean_plane_geometry Matroid_theory Theorems_in_discrete_geometry