Geometric constructibility of polygons lying on a circular arc

TL;DR

This paper proves that most polygons on a circle, called n-fans, cannot be constructed with straightedge and compass from their central angles and distances, especially for larger n and certain angles.

Contribution

It generalizes previous results by showing non-constructibility of n-fans for various angles and sizes, extending known special cases.

Findings

n-fans are generally not constructible from central angles and distances for n ≥ 3

For fixed δ in (0, 2π], some n-fans are explicitly non-constructible for n ≥ 5

Extends earlier results for special cases δ=2π and δ=π

Abstract

For a positive integer $n$, an $n$-sided polygon lying on a circular arc or, shortly, an $n$-fan is a sequence of $n+1$ points on a circle going counterclockwise such that the "total rotation" $\delta$ from the first point to the last one is at most $2\pi$. We prove that for $n\geq 3$, the $n$-fan cannot be constructed with straightedge and compass in general from its central angle $\delta$ and its central distances, which are the distances of the edges from the center of the circle. Also, we prove that for each fixed $\delta$ in the interval $(0, 2\pi]$ and for every $n\geq 5$, there exists a concrete $n$-fan with central angle $\delta$ that is not constructible from its central distances and $\delta$. The present paper generalizes some earlier results published by the second author and \'A. Kunos on the particular cases $\delta=2\pi$ and $\delta=\pi$.