# The conic geometry of rectangles inscribed in lines

## TL;DR

This paper explores the geometric relationships between rectangles inscribed in lines and their centers, using hyperbolic rotations and elliptical cones to develop a new theoretical framework.

## Contribution

It introduces a novel geometric approach connecting line configurations, elliptical cones, and rectangle centers, expanding understanding of inscribed rectangles in the plane.

## Key findings

- Established a link between inscribed rectangles and hyperbolic rotations.
- Derived properties of the locus of rectangle centers.
- Developed a new geometric framework involving elliptical cones.

## Abstract

We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.