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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 §4.3.4. 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 classes CL061 and CL062. See also Table 74, specially the pink cells in the table.

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.


K072 is invariant under both Cundy-Parry transformations defined in CL037.

K072 is spK(X542, X5) as in CL055. More generally, any spK(P on L∞, X5) is a focal nK(X6, R, X3) for some root R on the trilinear polar of X(264). Other examples are K072, K164, K165, K166, K187, K433, K932, K1154, K1180.