On the number of Euclidean ordinary points for lines in the plane

Given an arrangement of n not all coincident, not all parallel lines in the (projective or) Euclidean plane we have earlier shown that there must be at least (5n+6)/39 Euclidean ordinary points. We improve this result to n/6. 1 Sylvester’s problem in the Euclidean plane The classical Theorem of Sylvester and Gallai states that given a set of n not all collinear points in the plane, there must be at least one line which passes through exactly two of the points. The theorem has a corresponding dual statement, namely that any collection of n lines in the projective plane has at least one point where precisely two of the lines intersect. We call such a point an ordinary point. The Theorem of Sylvester and Galla