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X(4), X(5), X(13), X(14), X(30), X(79), X(80), X(265), X(621), X(622), X(1117), X(1141), X(5627), X(11584), X(14372), X(14373) extraversions of X(79) and X(80) Ixanticevian points : see table 23 points Na, Nb, Nc mentioned in Central Cubics 

Remember that the Neuberg cubic is the locus of point M such that the triangles ABC and MaMbMc are in perspective, where Ma, Mb, Mc are the reflections of M about the sidelines of ABC. This means that the lines AMa, BMb, CMc are concurrent at point N and then the locus of N is the cubic we call Kn. MaMbMc is called 2pedal triangle in Pinkernell's paper and Kn is the 2cevian cubic associated to the 2–pedal cubic (Neuberg cubic) and the –2–pedal cubic (Napoleon cubic). See the related quartic Q161 where a list of pairs {M, N} is given. Kn is the circular pivotal cubic with pivot X(265) = isogonal of inverse of H and pole a point named Po = X(1989) = isogonal of X(323). X(323) is the reflection of X(23) about X(110), where X(23) is the inverse of G and X(110) is the focus of the Kiepert parabola. Its singular focus (not on the curve) is X(3448), the reflection of X(110) about X(125) or the reflection of X(399) about X(5). Its orthic line is the Euler line. Kn is tangent at A, B, C to the altitudes of triangle ABC. Kn is tangent to the parallels to the Euler line at the in/excenters (those lines are also tangent to the Neuberg cubic), the contacts being X(79) and its extraversions. The fourth real intersection E with the circumcircle is the second intersection of the line X(5)X(110) with the circumcircle and also the sixth intersection with the rectangular hyperbola through A, B, C, H, X(5). This point is now called X(1141) in ETC. Its real asymptote is parallel to the Euler line at X(399) or X(323). Kn intersects its asymptote at Y = X(14451) on the line through X(265) and the antigonal of X(399), this latter point is X(1117) (labelled Le on the figure) lying on the Lester circle and on Kn. Kn is also the isogonal transform of K073, the inversive image of the Neuberg cubic in the circumcircle. In other words, Kn is the antigonal transform of the Neuberg cubic. Hence the antigonal of each point on the Neuberg cubic is a point on Kn and vice versa. Kn is a member of the class CL024 of cubics. It is anharmonically equivalent to the Neuberg cubic. See Table 20. The isotomic transform of Kn is K636, its anticomplement is K753 and its complement is K900. K060 and Q155 are both globally invariant under the inversion with pole X(79) which swaps X(80) and X(265). 


