Invariant Center Power and Elliptic Loci of Poncelet Triangles

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 linear combination of barycenter and circumcenter, its locus over the family is an ellipse.

Loci of Poncelet Triangles in the General Closure Case

“…Experiments suggests that only in the confocal pair 3 can the locus of the incenter be a conic [14].…”