The locations of triangle centers

Abstract. The orthocentroidal circle of a non-equilateral triangle has diameter GH where G is the centroid and H is the orthocenter. We show that the Fermat, Gergonne and symmedian points are confined to, and range freely over the inte-rior disk punctured at its center. The Mittenpunkt is also confined to and ranges freely over another punctured disk, and the second Fermat point is confined to and ranges freely over the exterior of the orthocentroidal circle. We also show that the circumcenter, centroid and symmedian point determine the sides of the reference triangle ABC. 1