Conic locus of inversive Poncelet circumcenter and two points of invariant circle power

TL;DR

This paper proves that the circumcenter of an inversive triangle in a Poncelet family traces a conic and confirms the existence of two points with constant power relative to key circles.

Contribution

It establishes the conic locus of the inversive triangle's circumcenter and verifies a conjecture about points with invariant power in Poncelet triangles.

Findings

Circumcenter locus is a conic in a Poncelet triangle family.

Two points with constant power relative to circumcircle and Euler's circle exist.

Confirmed an earlier conjecture about invariant power points.

Abstract

We prove that over a generic Poncelet triangle family, the locus of the circumcenter of an inversive triangle is a conic. Additionally, we prove an earlier conjecture: over generic Poncelet triangles, two unique points exist which maintain constant power with respect to the circumcircle and Euler's circle of the family, respectively.