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X(2), X(3), X(20), X(63), X(69), X(77), X(78), X(271), X(394) extraversions of X(63), X(77), X(78), X(271) 

Denote by A', B', C' the antipodes of A, B, C on the circumcircle. For any point P, Pa is the trace of PA' on BC, Pb, Pc similarly. The triangles ABC and PaPbPc are perspective if and only if P lies on the Darboux cubic. The locus of the perspector is the Darboux perspector cubic i.e. K099 = pK(X394, X69). K099 is anharmonically equivalent to the Thomson cubic. See Table 21. This cubic is a member of the class CL042. The isogonal transform of K099 is K445 = pK(X2207, X4) and its isotomic transform is K647 = pK(X2052, X264). 
