Before revision:  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:  For the confocal pair (elliptic billiard), we show that if the triangle 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.