In the plane of any triangle ABC, let G be the centroid, and let
La = the bisector of angle A,
Ga = reflection of line AG about La,
The three lines Ga, Gb, Gc meet in the symmedian point, labeled K in the sketch. It is also known as the Lemoine point. For a particularly good discussion, see
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, Washington, D.C., 1995.