On Spaces of Inscribed Triangles

TL;DR

This paper extends Meyerson's Theorem by demonstrating that for any Jordan loop, there are uncountably many inscribed triangles of various shapes, derived from limits of inscribed polygons.

Contribution

It introduces a new generalization of Meyerson's Theorem, showing the abundance of inscribed triangles of different shapes in Jordan loops.

Findings

Uncountably many inscribed triangles of various shapes exist in Jordan loops.

The result is obtained through limits of inscribed polygons.

Generalizes Meyerson's Theorem to broader classes of triangles.

Abstract

Meyerson's Theorem says that all but at most 2 points of any Jordan loop are vertices of inscribed equilateral triangles. We show that for any Jordan loop there are uncountable many other triangle shapes for which this same result is true. Our result comes from taking the limit of a structural result about spaces of triangles inscribed in polygons.