On Dirac and Motzkin problem in discrete geometry

TL;DR

This paper proves a special case of the Dirac and Motzkin conjecture, showing that for points on three concurrent lines, there exists a point incident with at least half of the lines spanned by the set.

Contribution

It establishes the conjecture for point sets distributed on three lines passing through a common point, a case not previously proven.

Findings

Any such set has a point incident with at least half the lines

Supports the conjecture in a specific geometric configuration

Advances understanding of incidences 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.