Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons

[2608.11208] Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > General Mathematics arXiv:2608.11208 (math) [Submitted on 27 Jun 2026 ( v1 ), last revised 14 Aug 2026 (this version, v2)] Title: Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons Authors: Mohammad Hassan Murad View a PDF of the paper titled Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons, by Mohammad Hassan Murad View PDF HTML (experimental) Abstract: The classical Blaschke-product approach provides an elegant description of triangles inscribed in a circle and circumscribed about an ellipse. Motivated by the Siebeck--Marden theorem, we derive a symmetric parametrization for cyclic $p$-gons circumscribed about a Siebeck--Marden curve of class $p-1$, recovering the Blaschke product parametrization as the triangular case. We then use this parametrization to establish several geometric invariance results, including a characterization of when the sum of the squares of all sides and diagonals is invariant. We revisit Cayley's criterion for 3- and 4-Poncelet pairs and obtain a complete classification of the associated central conics solely in terms of their foci. In particular, we show that the formulas for the lengths of the major axis obtained under the assumption on the foci of the conic lying inside the circumcircle remain valid when one focus or both foci lie outside the circumcircle, thereby extending the classical theory from ellipses to all admissible central conics. Several geometric properties of the associated cyclic quadrilaterals are also obtained. Comments: 27 pages, 7 figures Subjects: General Mathematics (math.GM) MSC classes: 2020: Primary: 51N20, Secondary: 30C15, 51M04 Cite as: arXiv:2608.11208 [math.GM] (or arXiv:2608.11208v2 [math.GM] for this version) https://doi.org/10.48550/arXiv.2608.11208 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mohammad Hassan Murad [ view email ] [v1] Sat, 27 Jun 2026 17:11:44 UTC (18 KB) [v2] Fri, 14 Aug 2026 21:55:41 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons, by Mohammad Hassan Murad View PDF HTML (experimental) TeX Source view license Current browse context: math.GM < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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