The wrought iron beauty of Poncelet loci

The paper states that over the confocal Poncelet triangle family, the loci of X1, X2, X3, X4 are ellipses, and points to external sources for formal proofs; it offers no proof in situ. This claim appears explicitly in the text and is illustrated (e.g., Fig. 7) and summarized as “Formal proofs appeared in [32,9,10]” for the first three centers . The candidate solution correctly reduces to 3-periodic orbits of the elliptic billiard and cites established results for incenter (Romaskevich), centroid (Garcia), circumcenter (Fierobe), and a general linear-combination theorem (Helman–Laurain–Garcia–Reznik) to deduce the orthocenter via H=3G−2O. The only gap in the candidate solution is a brief, non-rigorous justification that the locus equals the entire ellipse (not a proper subset); this can be remedied by citing explicit parametrizations or surjectivity arguments from the literature. Net: the paper is expository and incomplete proof-wise (especially for X4), while the model’s solution is substantively correct, given the cited dependencies.