Invariant Center Power and Elliptic Loci of Poncelet Triangles
This work addresses geometric properties in the context of Poncelet triangles, which is incremental for mathematical geometry research.
The paper tackles the problem of analyzing center power and triangle center loci for Poncelet triangles in nested ellipses, showing that the power of the center with respect to circumcircles or Euler's circles is invariant for concentric pairs, and that certain triangle centers trace elliptical loci.
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.