A Symmetric Polynomial Approach to Poncelet Triangles Inscribed in a circle and circumscribed about Central Conics
Year: 2026
Type: preprint
Abstract: We develop a symmetric-polynomial framework for the study of one-parameter families of Poncelet triangles. For triangles inscribed in the unit circle and circumscribed about a central conic, we show that the elementary symmetric polynomials of the vertices depend linearly on a single complex parameter. It follows that every symmetric rational function of the vertices is a rational function of this parameter, providing a unified algebraic approach to the investigation of geometric invariants. As applications, we recover several known invariance results and establish new ones involving angles, orthic triangles, tangential triangles, polar circles, and other classical constructions. We also obtain explicit formulas and loci for several associated geometric objects, leading to new invariance phenomena within Poncelet families.
Source: arXiv (Cornell University)
Author: Murad, Mohammad Hassan
Institution: The University of Texas at Arlington
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7147146902
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