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X(1), X(2), X(3), X(165), X(5373), X(6194), X(21214), X(32524)

excenters

infinite points of the McCay cubic K003

three contacts of the Steiner deltoid of IaIbIc with the circumcircle (vertices of the CircumTangential triangle, points on K024)

Q1, Q2, Q3 vertices of the Thomson triangle, on the circumcircle and on the Thomson cubic

other points and details below

Kp60 = Kp(X6) = K077 is the locus of pivots of isogonal pK+. The locus of the common point of the three asymptotes is Kc60 = Kc(X6) = K078.

This cubic is cited in several articles by R. Deaux, Mathesis 1959. See also the related cubic K609.

K078 is a member of the class CL015 of cubics.

K078 is a K60+ (a stelloid) with three real asymptotes parallel to the sidelines of the Morley triangle and concurring at X(3524), the homothetic of G under h(O,1/3). X(3524) is the centroid of the Thomson triangle.

K078 is the McCay cubic of triangle Q1Q2Q3. The tangents at these points and at O concur at G, the orthocenter of Q1Q2Q3. K078 is therefore invariant under isogonal conjugation with respect to the Thomson triangle. See the related K764, K765, K834 and also K1063, K1064.

K077, K078, K100, K258 are members of a same pencil of cubics generated by the McCay cubic K003 and the decomposed cubic which is the union of the Stammler hyperbola and the line at infinity. See Table 63.

The homothety h(O,-1/3) transforms K078 into K077. The reflection of K078 in O is K736.

The CT-isogonal transform of K078 is the central cubic K735.

Points on K078

K078 contains :

• the vertices T1T2T3 of the circumtangential triangle, these points lying on K024.

• the pedals G1, G2, G3 of G in the circumtangential triangle T1T2T3.

The lines GG1, GG2, GG3 are actually the asymptotes of the McCay cubic.

• the cevians O1, O2, O3 of O in the triangle Q1Q2Q3.

These points lie on the reflections about X(3524) of the asymptotes of the McCay cubic.

Note that the six points G1, G2, G3, O1, O2, O3 lie on a same conic which is the bicevian conic of X3 and X3524 in Q1Q2Q3.

• 6 points on the bisectors, homothetic of the intersections of the circumcircle with those bisectors in the homotheties with centre the corresponding vertex of ABC and ratio 2/3.

These points are two by two collinear with G.

• the third point on X(1)X(2) is X(21214), also on the analogous cubic K1063.

• the four foci of the conic with center X(3) inscribed in the Thomson triangle, see below.

Consider two conics C(P) and T(P) with same center P inscribed in the reference triangle ABC and in the Thomson triangle (T) respectively.

These conics have the same axes if and only if P lies on K078.

When P = X2, C(P) and T(P) are both the Steiner inellipse since it is inscribed in both triangles.

The figure is obtained when P = X(6194).

***

When P = X(3), these axes are parallel to the asymptotes of the Jerabek hyperbola (which is true for any point on the Stammler hyperbola) hence they are the lines through X(3) and X(2574), X(2575).

Since K078 is the McCay cubic of (T), the foci of T(X3) must lie on K078.

Note that these foci also lie on Q002 and K463, K804.

K078 generalized

Let (K1) = pK(X6, P), where P is not on the line at infinity. There is a cubic (K2) = pK(Ω, X4) sharing with (K1) the same points Q1, Q2, Q3 on the circumcircle (O). When P = u : v : w, this pole Ω is :

a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4 u-a^2 b^2 u-a^2 c^2 u+a^4 v-a^2 b^2 v+b^2 c^2 v-c^4 v+a^4 w-b^4 w-a^2 c^2 w+b^2 c^2 w) : : .

If M* denotes the isogonal conjugate of M with respect to triangle (T) = Q1Q2Q3, then the locus MC(P) of M such that O, M, M* are collinear is the McCay cubic of (T).

MC(P) obviously passes through O and also P, the orthocenter of (T) and the tangential of O.

For instance, with P = X(2), X(8), X(385), X(194), X(6), X(2979) we obtain the cubics K078, K1063, K1064, K1098, K1161, K1336 respectively.

Properties of MC(P) :

• MC(P) is a stelloid that meets the line at infinity at the same points as K003. It radial center X is the homothetic of P under h(O, 1/3). The six remaining common points lie on a same rectangular hyperbola H(P) passing through O.

H(P) decomposes into two secant lines (one of them passing through O) if and only if P lies on the anticomplement of K028. This is the case of K1098.

H(P) is a bicevian conic if and only if P lies on the anticomplement of K257 which is the isotomic transform of K028.

• MC(P) meets (O) at Q1, Q2, Q3 (with tangents passing through P) and at the vertices T1, T2, T3 of the CircumTangential triangle.

• MC(P) meets the sidelines of (T) again at the vertices O1, O2, O3 of the cevian triangle of O in (T).

• MC(P) meets the sidelines of T1T2T3 again at the vertices R1, R2, R3 of the pedal triangle of P with respect to T1T2T3.

• These six points O1, O2, O3, R1, R2, R3 lie on a same conic C(P) which is the bicevian conic C(O, X) in (T). C(P) also contains the midpoints M1, M2, M3 of (T). Its center Y is the homothetic of P under h(O, 1/4).

• MC(P) passes through the four foci of the conic with center O and inscribed in (T).

Equation of MC(P) :

b^4 c^2 v x^3 - b^2 c^4 v x^3 + b^4 c^2 w x^3 - b^2 c^4 w x^3 - b^4 c^2 u x^2 y + b^2 c^4 u x^2 y + 2 a^2 b^2 c^2 v x^2 y + a^2 b^2 c^2 w x^2 y - 2 a^2 b^2 c^2 u x y^2 + a^4 c^2 v x y^2 - a^2 c^4 v x y^2 - a^2 b^2 c^2 w x y^2 - a^4 c^2 u y^3 + a^2 c^4 u y^3 - a^4 c^2 w y^3 + a^2 c^4 w y^3 - b^4 c^2 u x^2 z + b^2 c^4 u x^2 z - a^2 b^2 c^2 v x^2 z - 2 a^2 b^2 c^2 w x^2 z + a^2 b^2 c^2 u y^2 z + a^4 c^2 v y^2 z - a^2 c^4 v y^2 z + 2 a^2 b^2 c^2 w y^2 z + 2 a^2 b^2 c^2 u x z^2 + a^2 b^2 c^2 v x z^2 - a^4 b^2 w x z^2 + a^2 b^4 w x z^2 - a^2 b^2 c^2 u y z^2 - 2 a^2 b^2 c^2 v y z^2 - a^4 b^2 w y z^2 + a^2 b^4 w y z^2 + a^4 b^2 u z^3 - a^2 b^4 u z^3 + a^4 b^2 v z^3 - a^2 b^4 v z^3 = 0