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X(1), X(2), X(6), X(31), X(32), X(869), X(985), X(2319) 

For any point M, let M' = GM /\ KM* (M* isogonal conjugate of M). Denote by (C) the circumconic with perspector a given point Q. M' lies on (C) if and only M lies on a nodal circumcubic with node K passing through G. This cubic is a K0 if and only if Q lies on the line GK. It meets the sidelines of ABC at three collinear points if and only if Q lies on the line at infinity, on the Lemoine axis or on the trilinear polar of X(83). In the former two cases, the cubic decomposes and in the latter case, it is a nK with pole X(32) passing through K and therefore a conicopivotal cubic with root a point on the line X(99)X(110). K285 is obtained with Q = X(659). The isogonal conjugate of K285 is K286. 
