Communicating harmonic pencils of lines

TL;DR

This paper investigates the properties of harmonic pencils of lines in the plane, exploring conditions for concurrency and collinearity, and generalizes classical theorems like Pappus, Desargues, Ceva, and Menelaos within a projective geometry framework.

Contribution

It introduces new conditions for harmonic pencils that lead to generalized versions of classical theorems, expanding the understanding of affine and projective configurations.

Findings

Conditions for concurrency and collinearity in harmonic pencils

Generalizations of Pappus, Desargues, Ceva, and Menelaos theorems

New perspectives on collinearity and concurrency in projective geometry

Abstract

Suppose there are $n$ harmonic pencils of lines given in the plane. We are interested in the question whether certain triples of these lines are concurrent or if triples of intersection points of these lines are collinear, provided that we impose suitable conditions on the initial harmonic pencils. Such conditions can be that certain of the given lines coincide, are concurrent or that certain intersection points are collinear. The study of these questions for $n=2, 3, 4$ sheds light on some well known affine configurations and provides new results in the projective setting. As applications, we will formulate generalizations or stronger versions of the theorems of Pappus, Desargues, Ceva and Menelaos. Notably, the generalized theorems of Ceva and Menelaos suggest a new way to generalize the terms collinearity and concurrency.