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X(2), X(3), X(4), X(6), X(542), X(842), X(6328), X(14246), X(14355), X(14356), X(14357), X(14366)

four foci of the MacBeath inconic : X(3), X(4) and two imaginary

This cubic is met in Special isocubics ยง4.3.4. It is the unique isogonal focal nK passing through G, O, H, K. See also K164 and the related K816.

It is an isogonal focal with singular focus X(842) and root X(9979), on the lines X(107)X(110), X(297)X(525), X(351)X(523). It is a member of the class CL061.

K072 is the locus of point M such that the segments GO and KH are seen from M under equal or supplementary angles. This shows that X(842) is the only point on the circumcircle having this property.

It is also the locus of point M whose pedal circle is centered on the orthic line which is the parallel at X(5) to X(2)X(98) or, equivalently, the locus of foci of inscribed conics centered on this parallel.

See Table 30 for other cubics of the same type.