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X(2), X(13), X(14), X(15), X(16), X(30), X(98), X(476), X(1316)

other points below

The union of the cevian lines of X(13) and the union of the cevian lines of X(14) can be considered as two (degenerate) pivotal equilateral cubics pK60++ i.e. with three real asymptotes concurring on the cubic. These cubics both contain A, B, C and the six other common points of these cevian lines, thus they generate a pencil of K60+ through these nine common points.

This pencil contains a third pK which is K037 and three nK among them K205.

Each cubic (K) of the pencil has three concurring asymptotes that meet at X on the circle through X(2), X(13), X(14), X(111), X(476). The hessian (H) of (K) is a focal cubic with singular focus X. Recall that the polar conic of each point P on (H) with respect to (K) is the union of two perpendicular lines that intersect at Q on (H) and vice versa. P and Q are said to be conjugated on (H).

The most remarkable hessian is K508, that with focus X(2), although the corresponding cubic (K) has very little interest apart the fact that it has three real concurring asymptotes making 60° angles with one another and meeting at G.

K508 is the locus of point M such that the directed line angles (MX13, MX15) and (MX16, MX14) or (MX13, MX16) and (MX15,MX14) are equal (mod. π). See the related property 1 in K018 and also K909.

Other points on K508 :

B3 = X(15), X(16) /\ X(476), X(1316), on the Brocard axis,

F3 = X(13), X(14) /\ X(98), B3, on the Fermat axis,

L3 = X(2), B3 /\ Lemoine axis (K508 meets the Lemoine axis again at two imaginary points that also lie on the circumcircle),

I3 = X(98), X(476) /\ X(30), L3 /\ X(1316), F3,

M3 = X(30), B3 /\ X(2), I3.

Conjugated points on K508 :

K508 is invariant under the Psi-involution hence, for any point P on K508, Q = Psi(P) is another point on the curve called the conjugate of P. The parallel at P to the Euler line meets K508 again at P' and then Q is the third point of K508 on the line X(2)P'.

K508 is a Fermat Psi-cubic as in Table 60 where the general case is studied.

It follows that the bisectors of any two lines GM, GN are the axes of the Steiner ellipses.

The polar conic of X(30) in K508 is the rectangular hyperbola passing through X(30), X(395), X(396), X(523) and the intersection of the lines X(6), X(523) and X(30), X(115).

This hyperbola meets the axes of the Steiner in-ellipse at four (not always real) points which are the centers of anallagmaty of K508.

The tangents at these four points are parallel to the real asymptote which is the line X(30)X(98).

When these points are all real, K508 contains the vertices of their diagonal triangle and K508 is an isogonal pK with pivot X(30) with respect to this triangle.

K508 is the locus of contacts of tangents drawn through G to the circles centered on X(6), X(523) and whose radical axis is the line X(30), X(115), the orthic line of the cubic.