Polygons inscribed in Jordan curves with prescribed edge ratios

TL;DR

This paper proves the existence of polygons with prescribed side ratios inscribed in smooth Jordan curves, extending classical results on inscribed polygons and similar triangles in geometric analysis.

Contribution

It establishes the existence of polygons with specified edge ratios inscribed in differentiable Jordan curves, generalizing the inscribed triangle problem.

Findings

Existence of polygons with prescribed edge ratios inscribed in Jordan curves.

Existence of similar triangles inscribed in Jordan curves.

Generalization of classical inscribed polygon problems.

Abstract

Let $J$ be a simple closed curve in $\mathbb R^{k}$ $(k\geq2)$ that is differentiable with non-zero derivative at a point $A_0\in J$. For a tuple of positive reals $a_1,\cdots,a_n$ $(n\geq3)$, each of which is less than the sum of the others, we show that there exists a polygon $Q_n$ inscribed in $J$ with sides of lengths proportional to $(a_1,\cdots,a_n)$. As a consequence, we prove the existence of triangle inscribed in $J$ similar to any given triangle.