On Dirac and Motzkin problem in discrete geometry

Abstract

Dirac and Motzkin conjectured that any set X of $n$ non-collinear points in the plane has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X. In this paper we prove that any set X of $n$ non-collinear points in the plane, distributed on three lines passing through a common point, has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X.