Before revision: We provide a theory as to why the locus of a triangle center over Poncelet 3-periodics in an ellipse pair is an ellipse or not. For the confocal pair (elliptic billiard), we show that if the  center can be expressed as a fixed affine combination of barycenter, circumcenter, and mittenpunkt (which is stationary over the confocal family) , then its locus will be an ellipse.
After revision: We provide a theory as to why the locus of a triangle center 's locus over a Poncelet 3-periodic family. We show that if the center can be expressed as a fixed affine combination of barycenter, circumcenter, and mittenpunkt (which is stationary over the confocal family) , then its locus will be an ellipse.