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X(3), X(4), X(468), X(895), X(3563), X(3564), X(6337)

X(3563) = singular focus,

X(3564) (infinite point of the line X(5)-K)

foci of the K-ellipse (inellipse with center K when the triangle ABC is acute angle)

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

Consider a point P, its isogonal conjugate P*, its reflection P# in the Euler line. The line P*P# passes through a fixed point Q on the Euler line if and only if P lies on an isogonal nK with root R on the trilinear polar of X(264), isotomic conjugate of the circumcenter O. All these cubics form a pencil of focal (van Rees) cubics with singular focus on the circumcircle. They all contain X(3) and X(4).

They are also :

  • the locus of P such that the segments OQ and HQ* are seen under equal or supplementary angles.
  • the locus of foci of inscribed conics centered on a line passing through X(5).

Among them, we find K072 obtained for Q = G, the strophoid K165 and K166 = Brocard (nineth) cubic when Q = X(1316). See the figure below.

This pencil contains one and only one nK0 which is K164 = van Lamoen cubic. It is obtained when Q = X(468), intersection of the Euler line and the orthic axis. (Floor van Lamoen, Hyacinthos #8382). It is the locus of foci of inscribed conics centered on the line passing through X(5) and X(6). The root of K164 is the point X(2501) = (b^2 - c^2) / SA : : on the orthic axis. Its singular focus is X(3563) on the lines G-X(136), H-X(99). K164 is a member of the class CL025. It is also spK(X3564, X6) in CL055.

More information about isogonal circular nKs in Special Isocubics ยง 4.1.2. See also the related K835.