X(3), X(4), X(30), X(74)
foci of the inscribed Steiner ellipse, see also Table 48.
foci of the inconic with center O, perspector X(69)
four foci of the MacBeath inconic : X(3), X(4) and two imaginary
feet of the trilinear polar of X(525)
K187= nK(X6, X525, X3) is an isogonal focal cubic with root X(525) and focus X(74).
K187 is the locus of foci of inscribed conics centered on the Euler line or, equivalently, the locus of M such that the midpoint of M and its isogonal conjugate gM lies on the Euler line. When "isogonal" is replaced with "isotomic", the locus is K953.
The real asymptote of K187 is parallel to the Euler line at X(110).
Let us consider the two following decomposed cubics : one is the union of the line at infinity and the Jerabek hyperbola, the other is the union of the circumcircle of ABC and the Euler line. Each one is clearly the isogonal transform of the other.
These cubics generate a pencil of circular circum-cubics passing through O, H, X(30), X(74). This pencil is stable under isogonal conjugation and contains the Neuberg cubic K001 (the only self-isogonal pK) and K187 (the only self-isogonal nK). The singular foci lie on the line O-X(74)-X(110)-etc and two isogonal conjugate cubics have their respective foci inverse in the circumcircle. The orthic line is the Euler line.