Natural Families of Triangles II: A Locus of Symmedian Points

We group triangles into families based on three parameters: the distance between the circumcenter O and the centroid G , the circumradius, and the measure of angle ∠GOA where A is one vertex. We focus on the family of triangles which allows ∠GOA to vary and fixes the other two parameters. By construction, this grouping produces triangles which share the same Euler line. Perhaps unexpectedly, if we examine the family’s locus of a triangle center known as the symmedian point, we find that it always forms an arc of a circle centered at a specified point on the Euler line.