# On the Triangles in Certain Types of Line Arrangements

## TL;DR

This paper provides a combinatorial framework to describe and distinguish triangles in specific line arrangements with cyclic or infinity properties, offering new characterizations and unique determination results.

## Contribution

It introduces a novel combinatorial nomenclature for line arrangements and proves properties that distinguish arrangements with cyclicity from infinity type arrangements.

## Key findings

- Triangles are characterized in both arrangement types.
- The set of triangles uniquely determines arrangements with cyclicity.
- Counterexamples show non-uniqueness for infinity arrangements.

## Abstract

In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been described for both the types and some properties of the nomenclature have been proved. Later using the nomenclature we describe the triangles present in both types of line arrangements in Theorems $A,B$. We also prove that the set of triangles uniquely determine, in a certain precise sense, the line arrangements with global cyclicity and not the infinity type line arrangements where counter examples have been provided. In Theorem $9.1$, given a nomenclature, we characterize when a particular line symbol in the nomenclature is a line at infinity for the arrangement determined by the nomenclature.