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Journals / International Electronic Journal of Geometry / 2023 / Cilt: 16 Sayı: 2

The Pentagon Theorem in Miquelian Möbius Planes

Lorenz HALBEISEN
Norbert HUNGERBÜHLER
Vanessa LOUREİRO
Pages
689–696
Publication Date
2023-10-29
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.

📌 3 Key Takeaways

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Objective & Hypothesis:We give an algebraic proof of the Pentagon Theorem.
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Key Findings & Data:The proof works in all Miquelian Möbius planes obtained from a separable quadratic field extension.
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Conclusion & Impact:In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Pentagon theorem · Möbius planes · Miquel planes

Abstract

We give an algebraic proof of the Pentagon Theorem. The proof works in all Miquelian Möbius planes obtained from a separable quadratic field extension. In particular, the theorem holds in every finite Miquelian plane. The arguments also reveal that the five concyclic points in the Pentagon Theorem are either pairwise distinct or identical to one single point. In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.

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Cite This Article

HALBEISEN, L., HUNGERBÜHLER, N., LOUREİRO, V. (2023). The Pentagon Theorem in Miquelian Möbius Planes. International Electronic Journal of Geometry, Cilt: 16 Sayı: 2, 689–696. https://ajindex.com/articles/971854

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References (7)

  1. Benz, W.: Über Möbiusebenen. Jahresbericht der Deutschen Mathematiker-Vereinigung. 63 (Abt. 1), 1–27 (1960)
  2. Chen, Y.: Der Satz von Miquel in der Möbiusebene. Mathematische Annalen. 186, 81–100 (1970)
  3. Dembowski, P.: Finite geometries. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44. Springer-Verlag, Berlin-New York (1968)
  4. Hering, C.: Eine Klassifikation der Möbius-Ebenen. Mathematische Zeitschrift. 87, 252–262 (1965)
  5. Krier, N.: The Hering classification of Möbius planes. In: Proceedings of the International Conference on Projective Planes, Washington State Univerity, Pullman, Wash., 1973. Washington State University Press, Pullman. 157–163 (1973)
  6. Li, H., Xu, R., Zhang, N.: On Miquel’s five-circle theorem. In: Hongbo Li, Peter J. Olver, and Gerald Sommer (eds). Computer Algebra and Geometric Algebra with Applications. Springer, Berlin Heidelberg. 217–228 (2005)
  7. Miquel, A.: Théorèmes de géométrie. Journal de Mathématiques Pures et Appliquées. 3, 485–487 (1838).