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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 K 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. |
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Notes • when P lies on the Euler line, K contains X(3), X(25). Every cubic meets the circumcircle at the same points as an isogonal pK with pivot on the line GK. See orange cells. • 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. |
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