# Triangle areas in line arrangements

## TL;DR

This paper investigates the maximum and minimum number of triangles with specific areas determined by arrangements of lines in the plane, providing bounds and connections to combinatorial and incidence geometry.

## Contribution

It establishes bounds for the maximum number of unit and minimum area triangles in line arrangements, linking geometric problems to additive combinatorics and incidence theorems.

## Key findings

- Maximum unit area triangles: between Ω(n^2) and O(n^{9/4})
- Maximum minimum area triangles: tight bounds established
- Bounds for maximum and distinct area triangles using algebraic, geometric, and combinatorial methods

## Abstract

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.