Elliptic Billiards and Ellipses Associated to the 3-Periodic Orbits

The centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses. In this article, we obtain the canonical equations of these ellipses. Moreover, we complement the previous results about incenters obtained by Romaskevich. Also the geometric locus defined by barycenters of the triangles of an elliptic billiard are ellipses as established by Schwartz and Tabachnikov and explicit equations of these ellipses are also given. In addition, we obtain that the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse and the canonical equation is obtained.