Invariant Center Power and Elliptic Loci of Poncelet Triangles

TL;DR

This paper investigates the invariance of center power with respect to certain circles in Poncelet triangles and characterizes the loci of specific triangle centers as ellipses, revealing new geometric invariants and locus shapes.

Contribution

It establishes invariance of center power for concentric ellipse pairs and describes the elliptical loci of certain affine combinations of triangle centers in Poncelet 3-periodics.

Findings

Center power with respect to circumcircle or Euler's circle is invariant for concentric pairs.

Loci of specific triangle centers form ellipses in generic nested pairs.

Affine combinations of barycenter and circumcenter have elliptical loci.

Abstract

We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler's circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed affine combination of barycenter and circumcenter, its locus over the family is an ellipse.