  A triangle center such as the incenter, barycenter, etc., is specified by a
function thrice- and cyclically applied on sidelengths and/or angles. Consider
the 1d family of 3-periodics in the elliptic billiard, and the loci of its
triangle centers. Some will sweep ellipses, and others higher-degree algebraic
curves. We propose two rigorous methods to prove if the locus of a given center
is an ellipse: one based on computer algebra, and another based on an
algebro-geometric method. We also prove that if the triangle center function is
rational on sidelengths, the locus is algebraic
