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The circumcevian and circumanticevian triangles of any point M are perspective at Q. See Table 6. Now, if P is a fixed point, the points P, M, Q are collinear if and only if M lies on the pivotal cubic with pole X(32) and pivot P. All pivotal cubics with pole X(32) contain the Lemoine point X(6) and the vertices of the tangential triangle KaKbKc. The isogonal transform K* of K = pK(X32, P) is pK(X2, tgP), the isotomic pivotal pK with pivot tgP (the isotomic conjugate of the isogonal conjugate of P). See the related Table 35 (for cells highlighted in orange, P on the Euler line) and CL048 for locus properties. Any cubic K = pK(X32, P) is the barycentric product by X(1) of a cubic K' = pK(X6, P') where P' = P ÷ X(1) = P x X(75). In other words, the trilinear equation of K is the barycentric equation of K'. These three cubics K, K*, K' are equivalent. The following table shows a large selection of these pKs with at least eight ETC centers and also several special cubics. See notes below table. 



Notes • when P lies on the line {1, 19}, K contains X(19), X(48) and K' is an Euler cubic of Table 27. See green cells. • when P lies on the line {1, 3}, K contains X(55), X(56). See blue cells. • when P lies on the Euler line, K contains X(3), X(25). Each cubic meets the circumcircle at the same points T1, T2, T3 as an isogonal pK whose pivot lies on the line GK and the line at infinity at the same points as an isogonal pK whose pivot lies on the line {66, 69, ...}. See orange cells. Moreover, the sidelines of T1T2T3 envelope the parabola with focus X(111) and directrix the line GK. The conic inscribed in both triangles ABC and T1T2T3 has its center on GK. These two conics are tangent to the trilinear polar of X(523) passing through X(115) and X(125). The isogonal conjugation with respect to T1T2T3 transforms K into its adjunct central cubic K" as in CL076. The infinite points of K" are those of K004 and the center X lies on the line {2, 98, 110, etc}. See K1363 for instance.

