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X(30)

X = X(5627) = isogonal of X(1511) = X(1989)÷X(30)

K139 is an example of nK60++ cubic i.e. equilateral central non-pivotal isocubic with center X(5627). See Special Isocubics §7.4.2.

K139 is a stelloidal nK with pole X(1989) and root P = X(14566) = X(2)X(525) /\ X(5)X(523).

The asymptotes concur at X, the isogonal conjugate X(5627) of X(1511). One of them is the line X(30)X(74) – which is the real asymptote of the Neuberg cubic – and the two other are obtained by rotations about X with angles ±60°. These two asymptotes are parallel to those of (H), the circumconic with perspector X(1989), a hyperbola with eccentricity 2.

K139 is the locus of M such that the midpoint of MM* lies on the line X(30)X(74) where M* denotes the X(1989)–isoconjugate of M. Recall that X(1989) is the pole of the isoconjugation that swaps the Fermat points X(13) and X(14).

For any point M on K139, its reflection M' about X lies on the cubic and the line X(30)M' contains M*.