A Constructive Approach to Sylvester's Conjecture

Sylvester's conjecture states that, given n distinct noncollinear points in a plane, there exists a connecting line of two of the points such that no other point is incident with the line. First a proof is given of the six-point Sylvester conjecture from a constructive axiomatization of plane incidence geometry. Next ordering principles are studied that are needed for the seven-point case. This results in a symmetrically ordered plane affine geometry. A corollary is the axiom of complete quadrangles. Finally, it is shown that the problem admits of an arithmetic translation by which Sylvester's conjcture is decidable for any n.