Paths of rectangles inscribed in lines over fields

TL;DR

This paper explores the geometric paths of rectangles inscribed in lines over arbitrary fields, using algebraic methods to analyze how these paths depend on line configurations and parameters like slope and aspect ratio.

Contribution

It introduces algebraic techniques to study the geometry of inscribed rectangles parametrized by slope and aspect ratio over any field.

Findings

Describes the geometry of inscribed rectangle paths based on line configurations

Develops algebraic methods applicable over arbitrary fields

Analyzes dependence of paths on four-line arrangements

Abstract

We study rectangles inscribed in lines in the plane by parametrizing these rectangles in two ways, one involving slope and the other aspect ratio. This produces two paths, one that finds rectangles with specified slope and the other rectangles with specified aspect ratio. We describe the geometry of these paths and its dependence on the choice of four lines. Our methods are algebraic and work over an arbitrary field.