A Theory for Locus Ellipticity of Poncelet 3-Periodic Centers

We present a theory which predicts when the locus of a triangle center is an ellipse over a Poncelet family of triangles: this happens if the triangle center is a fixed affine combination of barycenter, circumcenter, and a third center which remains stationary over the family. We verify the theory works for the confocal and "with incircle" Poncelet families. For the confocal case, we also derive conditions under which a locus degenerates to a segment or is a circle. We show a locus turning number is either plus or minus 3 and predict its movement monotonicity with respect to vertices of the family.