Given an isoconjugation with pole W(p : q : r), the locus of point P such that the pedal triangles of P and its W-isoconjugate P* are parallelogic is nK0(W, X6), if we omit the line at infinity and the trilinear polar of the isotomic conjugate of the isogonal conjugate of W. See another generalization at K024 property 11. All these cubics contain the centers of the 3 Apollonian circles.
Compare this type of cubic with CL021.
The barycentric equation of nK0(W, X6) is : ∑ a^2 x (r y^2 + q z^2) = 0.
Each nK0(W, K) can also be seen as the locus of point P such that P and P* are conjugated with regard to the circumcircle.
More generally, each nK0(W, R) is the locus of point P such that P and P* are conjugated with regard to the circum-conic with perspector R.
The following table gives a selection of nK0(W, K) already mentioned in CTC.