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∑ a^4 (a^2 b^2+a^2 c^2-b^4-c^4) x (c^4y^2-b^4z^2) = 0

X(3), X(6), X(25), X(98), X(237), X(694), X(1691), X(1971), X(1987), X(3511), X(46272), X(61098), X(61099), X(61100)

X(61098), X(61099) : isogonal conjugates of X(25332), X(39355)

vertices of the cevian triangle of X(237)

vertices of the anticevian triangle of X(6)

imaginary points on (O) and Lemoine axis

Geometric properties :

K1363 = pK(X32, X237) is the isogonal transform of K355 = pK(X2, X511). See Table 34 for analogous cubics.

K380 is the locus of pivots of pKs meeting the circumcircle at the same points as pK(X6, X385) = K128. These points are the Tarry point X(98) and two imaginary points O1, O2 on the Lemoines axis. These points are in fact the isogonal conjugates of the infinite points of the Steiner ellipses. The locus of the poles is pK(X32 x X1976, X1976) and the locus of the isopivots is K1363.

K1363 meets the line at infinity at the same points as the isogonal pK with pivot P = a^2 (a^4 b^4-a^2 b^6+a^4 b^2 c^2-a^2 b^4 c^2+a^4 c^4-a^2 b^2 c^4+b^4 c^4-a^2 c^6) : : , SEARCH = -2.74709235867492, on the lines {66, 69}, {99, 2387}, {110, 1971}, {147, 511}, etc. P is now X(61101) in ETC.

Note that the isopivot of K1363 is X(98) hence, for any point M on the cubic, the X(237)-Ceva conjugate N of M is also on the cubic and X(98), M, N are collinear.

For instance, with M = X(25), we find N = P1 = X(61100).

The isogonal conjugation in the improper triangle with vertices X(98), O1, O2 transforms K1363 into its adjunct central cubic as in CL076.

Its center is X(98) and it passes through the infinite points of the altitudes of ABC, O1 and O2, and the centers X(4), X(98), X(385), X(5999), X(9862), X(11676), also the reflection Q of X(11676) in X(4).

The remaining points on (O) are the antipodes of the points (apart A, B, C) where pK(X6, P) meets (O).

Q = a^8+2 a^6 b^2-a^4 b^4-2 a^2 b^6+2 a^6 c^2-3 a^4 b^2 c^2+2 a^2 b^4 c^2-3 b^6 c^2-a^4 c^4+2 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-3 b^2 c^6 : : , SEARCH = 34.0597564794179, on the lines {4, 6}, {30, 148}, {74, 290}, {76, 3098}, {98, 187}, etc.

Q is now X(61102) in ETC.