Rational barycentric coordinates imply elliptic triangle-center loci

Let a triangle center be defined by barycentric coordinates that are rational functions of the squares of the triangle's sidelengths, and consider its locus over the relevant family of triangles. Ellipticity conjecture. The locus of the triangle center will be an ellipse. This claim is motivated by experimental observations of elliptic loci for triangle centers over concentric, axis-parallel families; a general theoretical explanation remains open.