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X(3), X(15), X(16), X(35), X(36), X(54), X(74), X(186), X(1154), X(1511), X(3165), X(3166), X(3438), X(3439), X(6104), X(6105), X(14354), X(14367), X(14368), X(14369), i X(370), ig X(370)

isogonal conjugates of the Ix-anticevian points. See Table 23.

Ki is the inversive image of the Neuberg cubic in the circumcircle and the isogonal transform of the Kn cubic. See also Inverses of Isocubics.

It is a circular pivotal cubic with pivot O = X(3) and pole X(50). The tangent at O is the Euler line. The singular focus is X(14652), the inverse of X(3448) in the circumcircle.

It is anharmonically equivalent to the Neuberg cubic. See Table 20.

K073 contains the midpoints of the segments joigning O and the three common points of the circumcircle and the Napoleon cubic. See a figure at Q011.

Locus properties :

  1. Ki is the locus of point P such that the pedal triangles of P and its X(50)-isoconjugate P* are orthologic. It is the only circular cubic of this type. See CL021.
  2. Denote by A'B'C' the reflection triangle of P in the sidelines of ABC and by A", B", C" the reflections of A, B, C in the sidelines of A'B'C'. The triangles A'B'C' and A"B"C" are perspective (at Q) if and only if P lies on Ki (Paul Yiu, Hyacinthos #8631). The locus of Q is Q110.
  3. Locus of isopivots of circular pKs which pass through the isodynamic points X(15), X(16). The locus of the poles is pK(X6 x X50, X6) and the locus of the pivots is K001.