Conic Loci of Poncelet Triangle Centers via Caustic Geometry — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22805957
Primary Topic
Mathematics and Applications
Type
preprint
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preprint

Conic Loci of Poncelet Triangle Centers via Caustic Geometry — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Mathematics and Applications
preprint

Conic Loci of Poncelet Triangle Centers via Caustic Geometry — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Poncelet triangle families yield conic loci for triangle centers, with a theory predicting ellipticity based on caustic geometry and trilinear polarity. | MATH: Poncelet porism: if one triangle is inscribed in conic E and circumscribed about conic K, then infinitely many such triangles exist (closure condition). For circular caustic K=(C,r), the locus of a triangle center X(n) is a conic iff a certain invariant (related to the center's barycentric/trilinear coordinates) satisfies a quadratic condition. The "orthofocal" circumellipse has a focus at X(4) (orthocenter); its center is a specific point (likely X(110) or a derived center). Inparabola focus lies on the circumcircle — a known theorem (focus of any inscribed parabola is on the circumcircle). The theory in arXiv:2106.00715v4 gives a criterion: for confocal pairs (E, inner confocal ellipse), the locus is a conic for centers whose coordinates satisfy a linear relation in the elliptic modulus; for concentric circular caust Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Mathematics and Applications
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Conic Loci of Poncelet Triangle Centers via Caustic Geometry — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS