Abstract
We study convex cyclic polygons, that is, inscribed n-gons. Starting from P. Schreiber’s idea, published in 1993, we prove that these polygons are not constructible from their side lengths with straightedge and compass, provided n is at least five. They are non-constructible even in the particular case where they only have two different integer side lengths, provided that n ≠ 6. To achieve this goal, we develop two tools of separate interest. First, we prove a limit theorem stating that, under reasonable conditions, geometric constructibility is preserved under taking limits. To do so, we tailor a particular case of Puiseux’s classical theorem on some generalized power series, called Puiseux series, over algebraically closed fields to an analogous theorem on these series over real square root closed fields. Second, based on Hilbert’s irreducibility theorem, we give a rational parameter theorem that, under reasonable conditions again, turns a non-constructibility result with a transcendental parameter into a non-constructibility result with a rational parameter. For n even and at least six, we give an elementary proof for the non-constructibility of the cyclic n-gon from its side lengths and, also, from the distances of its sides from the center of the circumscribed circle. The fact that the cyclic n-gon is constructible from these distances for n = 4 but non-constructible for n = 3 exemplifies that some conditions of the limit theorem cannot be omitted.
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Dedicated to the eightieth birthday of Professor László Leindler
Communicated by Á. Kurusa
This research was supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences” of project number “TÁMOP-4.2.2.A-11/1/KONV-2012-0073”, and by NFSR of Hungary (OTKA), grant number K83219.
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Czédli, G., Kunos, Á. Geometric constructibility of cyclic polygons and a limit theorem. ActaSci.Math. 81, 643–683 (2015). https://doi.org/10.14232/actasm-015-259-3
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DOI: https://doi.org/10.14232/actasm-015-259-3
Key words and phrases
- inscribed polygon
- cyclic polygon
- circumscribed polygon
- compass and ruler
- straightedge and compass
- geometric constructibility
- Puiseux series
- power series
- holomorphic function
- field extension
- Hilbert’s irreducibility theorem