Before revision:   Here we present a systematic method to prove 29 out of the first 100 Centers listed in Clark Kimberling's Encyclopedia are elliptic. We also derive conditions under which loci are algebraic .
After revision:  Here we present two rigorous methods to detect when any given Center produces an elliptic locus: a first one which is a hybrid of numeric and computer algebra techniques (good for fast detection only), and a second one based on the Theory of Resultants, which computes the implicit two-variable polynomial whose zero set contains the locus .