∑ (b^2 + c^2) x^2 (c^2y - b^2 z) = 0
X(2), X(4), X(6), X(83), X(251), X(1176), X(1342), X(1343)
vertices G1, G2, G3 of the Grebe triangle
infinite points of pK(X6, X141)
A'B'C' cevian triangle of X(83)
F, F' foci of the in-conic with perspector X(83), center X(3589)
K644 is another example of cubic passing through the vertices G1, G2, G3 of the Grebe triangle and here, the tangents are concurrent at X on the lines X(2)X(1285) and X(83)X(183) with first barycentric 5 a^4+7 a^2 b^2+7 a^2 c^2+8 b^2 c^2 and SEARCH = 2.07253149618850. X is now X(14535) in ETC (2017-09-25).
The tangents at A', B', C', K pass through G and the tangents at A, B, C, X(83) pass through K. The tangent at G is the Euler line.
K644 meets the in-conic with perspector X(83) at A', B', C' and three other points which are the contacts of this conic with the sidelines of the Grebe triangle. In other words, this conic is inscribed in ABC and in G1G2G3.
The isogonal transform of K644 is K836 = pK(X39, X2).
K644 is transformed into K1241 under the isogonal conjugation with respect to the Grebe triangle.
Note that the two isogonal conjugates of M on K644 lie on a line passing through O.