Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

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 focal 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 A, B, C, X(3), X(4), the imaginary foci of the MacBeath inconic and the circular points at infinity. See Table 80, MacBeath cubics. The real infinite point is that of the line {X110, Q} and its isogonal conjugate on the circumcircle is the singular focus of the cubic. O and H share the same tangential, namely Q*.

These nKs are also :

• the locus of P such that the directed angles (PO, PQ) and (PH, PQ*) are opposite or supplementary.
• the locus of foci of inscribed conics centered on a line passing through X(5).

Every cubic is invariant under the two Cundy-Parry transformations. See CL037.

Among them, we find K072 when Q = G, the strophoid K165 when Q = X(3109), 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 § 4.1.2. See also the related K835. See Table 73 for properties and other related cubics.