If R is different of G and K, there is another point R' such that the two corresponding triangles have the same circumradius r : R' is the harmonic conjugate of R with respect to G and K.
In figure 1, R = X(323) and r is the radius of the circumcircle of ABC.
In figure 2, R and R' are not mentioned in the current edition of Clark Kimberling's ETC. Their barycentric coordinates are : 2 a^2(a+b+c) ± 3 abc : : . In both cases, r is the radius of the incircle.
Other examples of such cubics are K085, K098, K105, K686.