centers of Apollonian circles
Given the isoconjugation with pole W = X110^2 (barycentric square of X(110), the locus of point P such that the pedal triangles of P and its W-isoconjugate P* are parallelogic is K147 = nK0(W, X6) = cK0(#X110, X6), if we omit the line at infinity and the trilinear polar of the isotomic conjugate of the isogonal conjugate of W. K147 is a member of the class CL022 of cubics.
It is a singular cubic with node X(110) and with perpendicular nodal tangents. See Special Isocubics §8.
The isogonal transform of K147 is K979.
See here for a family of related cubics.