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X(3), X(15), X(16), X(98), X(385)

points at infinity of K003

vertices of the CircumTangential triangle

other points below

Geometric properties :

Consider the imaginary triangle (T) whose vertices are X(98) and I1, I2, the common (always complex conjugated) points on the circumcircle (O) and the Lemoine axis. These two points are the isogonal conjugates of the infinite points of the Steiner ellipses. They are the common points – apart A, B, C – of K128 = pK(X6, X385) and (O).

Although (T) is imaginary, an isogonal conjugation can be defined with respect to (T) and then, K1064 is the McCay cubic K003 for this triangle. See Table 62 for other non proper triangles and the analogous cubics K078, K1063.

K1064 is a stelloid with three real asymptotes parallel to those of K003 and concurring at the radial center X, the homothetic of X(385) under h(X3, 1/3), on the lines {3,194}, {4,230}, {32,262}, {98,187}, {140,2896}, etc. The six remaining (finite) common points lie on the rectangular hyperbola passing through X(3), X(182), X(1976), X(2909), X(3148).

K1064 meets (O) at X(98), I1, I2 and the vertices of the CircumTangential triangle on K024.

Apart I1 and I2, K1064 meets the Lemoine axis again at P1 = a^4 (a^4 b^4-a^2 b^6-a^2 b^4 c^2+a^4 c^4-a^2 b^2 c^4+2 b^4 c^4-a^2 c^6) : : , SEARCH = 0.958796600855261, on the lines {3,76}, {32,8789}, {111,11332}, {187,237}, {446,2794}, etc.

P1 = X(21444) and X = X(21445) in ETC now (2018-08-24).