Abstract
In 1993 Watkins and Zeitlin published a method [1] to simply compute the minimal polynomial of \(\cos (2\pi /n)\), based on the Chebyshev polynomials of the first kind. For the work presented in this paper we have implemented a small augmentation, based on Watkins an Zeitlin’s work, to the dynamic mathematics tool GeoGebra. We show that this improves GeoGebra’s capability to discover and automatically prove various non-trivial properties of regular n-gons.
Discovering and proving, in this context, means that the user can sketch a conjecture by drawing the geometric figure with the tools provided by GeoGebra. Then, even if the construction is just approximate, in the background a rigorous proof is computed, ensuring that the conjecture can be confirmed, or must be rejected.
In this paper the potential interest of automated reasoning tools will be illustrated, by describing such new results in detail, obtained by some recently implemented features in GeoGebra.
Besides confirming well known results, many interesting new theorems can be found, including statements on a regular 11-gon that are impossible to represent with classical means, for example, with a compass and a straightedge, or with origami.
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Acknowledgments
The author was partially supported by a grant MTM2017-88796-P from the Spanish MINECO (Ministerio de Economia y Competitividad) and the ERDF (European Regional Development Fund).
Many thanks to Tomás Recio and Francisco Botana for their valuable comments and suggestions.
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Kovács, Z. (2018). Finding and Proving New Geometry Theorems in Regular Polygons with Dynamic Geometry and Automated Reasoning Tools. In: Rabe, F., Farmer, W., Passmore, G., Youssef, A. (eds) Intelligent Computer Mathematics. CICM 2018. Lecture Notes in Computer Science(), vol 11006. Springer, Cham. https://doi.org/10.1007/978-3-319-96812-4_15
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DOI: https://doi.org/10.1007/978-3-319-96812-4_15
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