Geometric constructibility of polygons lying on a circular arc

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 $\delta=\pi$.