G, K, and the Grebe cubic
The Grebe cubic is the locus of points P such that the center S of the bicevian conic of P and its isogonal conjugate lies on line PP*. In the particular case of P=K (symmedian point) and P*=G (centroid) we have always the ratio GS:SK=1:3.

If ABC is a triangle and P, Q are two points, the six traces of these points (the intersection points of lines AP, BP, CP, AQ, BQ, CQ with lines BC, CA, AB, BC, CA, AB, respectively) are always on a conic, that we call the bicevian conic of P and Q, and we denote as G(P, Q).

In the particular case when P=K (the symmedian point of ABC) and Q=G (the centroid of ABC), we the following results:

  1. The bicevian conic G(K, G) is always an ellipse.
  2. The center S of G(K, G) lies on line KG.
  3. The points G, S, K satisfy the ratio GS:SK = 1: 3.

If we ask for the isogonal conjugate pairs P, P* (from which K,G is an example) such that P, P* and the center S of the bicevian conic G(P,P*) are collinear we get the Grebe cubic, catalogued as K102 by Bernard Gibert.