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X(110), X(691), X(4230), X(5467), X(17938), X(17939), X(17940), X(17941), X(17942), X(17943), X(17944), X(17945), X(23348) centers of Apollonius 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 Wisoconjugate 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 ยง8. The isogonal transform of K147 is K979. K147 is the X(110)Hirst inverse of the circumcircle (O) of ABC. See here for a family of related cubics. W is now X(23357) in ETC (20180919). It is the isogonal conjugate of X(338). 
