Triangles Circumscribed about Central Conics and Their Invariants

TL;DR

This paper explores geometric invariants and explicit constructions of triangles inscribed in a circle and circumscribed about central conics, extending classical relations within Poncelet geometry.

Contribution

It introduces new invariants and explicit constructions linking triangle centers, central conics, and Poncelet families, expanding classical triangle geometry.

Findings

Identified invariants when the circumcenter coincides with conic centers or foci.

Derived explicit constructions of central conics related to specific triangle configurations.

Analyzed extremal area problems and sequences of Poncelet pairs within the geometric framework.

Abstract

We study families of triangles that are inscribed in a fixed circle and circumscribed about a central conic, extending the classical Chapple--Euler relation within the framework of Poncelet geometry. We establish several geometric invariants that arise when the circumcenter of the triangle coincides with either the center of the conic or one of its foci. These include invariance properties of orthic triangles, tangential triangles, polar circles, and trigonometric expressions such as $\sin^2 A + \sin^2 B + \sin^2 C$. We further derive explicit analytic constructions of central conics associated with a given triangle under special configurations, including cases where the foci are located at the circumcenter and orthocenter. In addition, we investigate extremal area problem within Poncelet families, and develop both homothetic and non-homothetic constructions of sequences of Poncelet pairs. These results provide a unified geometric framework linking classical triangle geometry, central conics, and Poncelet porism.