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K459

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X(5), X(39), X(395), X(396), X(511), X(597), X(3413), X(3414)

X(3413), X(3414) : infinite points of the Kiepert hyperbola

For any point M on the Brocard axis, the isogonal conjugate M* of M lies on the Kiepert hyperbola and the locus of the midpoint of MM* is the Brocard-Kiepert midpoint cubic K459.

K459 is an acnonal cubic with node X(39), the Brocard midpoint. The three real inflexion points are X(395), X(396), X(597).

K459 has three real asymptotes : two are parallel at X(620) to those of the Kiepert hyperbola and one is parallel to the Brocard axis.

Compare K459 and K460.