A tetrahedron is called equifacial (or isosceles) if all its 2-faces are congruent triangles. Many characterizations of equifacial tetrahedra are known. Some of these characterizations use the coincidence of special points usually called centers of the tetrahedron [see N. Altshiller-Court, “Modern Pure Solid Geometry”, 2nd ed. (Chelsea Publ. Comp., Bronx, NY) (1964; Zbl 0126.16603)].
In the paper under review, the authors present a nice proof of the statement that if the incenter and Fermat-Torricelli center (the unique point having minimal distance sum to the vertices) of a non-degenerate tetrahedron \(T\) coincide, then \(T\) is equifacial. It follows from this and Theorem 4 of M. Majja and P. Walker [Int. J. Math. Educ. Sci. Technol. 32, No. 4, 501–508 (2001; Zbl 1011.51007)] that a tetrahedron is equifacial if and only if any two of the incenter, the circumcenter, the centroid, and the Fermat-Torricelli center coincide.
In higher dimension, the various degrees of regularity implied by the coincidence of two or more centers of a simplex are investigated by A. L. Edmonds, M. Hajja, and H. Martini in [Beiträge Algebra Geom. 46, 491–512 (2005; Zbl 1093.51014) and Results Math. 47, 266–295 (2005; Zbl 1084.51008)]. In particular, it is shown that if \(d> 3\), then the coincidence of any two of the traditional centers of a \(d\)-simplex does not imply the equifaciality of the simplex, while the coincidence of any two such centers for an orthocentric \(d\)-simplex (a simplex whose altitudes intersect in a common point) implies regularity.