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 \(n\ge 3\), 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 \(n\ge 5\), 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 Á. Kunos on the particular cases \(\delta =2\pi \) and \(\delta =\pi \).
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This research of the second and third authors is supported by the Hungarian Research Grant KH 126581.
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Ahmed, D., Czédli, G. & Horváth, E.K. Geometric Constructibility of Polygons Lying on a Circular Arc. Mediterr. J. Math. 15, 133 (2018). https://doi.org/10.1007/s00009-018-1166-0
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DOI: https://doi.org/10.1007/s00009-018-1166-0