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X(4), X(6), X(15), X(16), X(524) A', B', C' : reflections of A, B, C in the sidelines of ABC A1, B1, C1 : second common points of the symmedians and the circumcircle A2 homothetic of A1 under h(A,1/2), B2, C2 similarly foci of the inscribed Steiner ellipse. 

K = X(6) is the Lemoine point of triangle ABC. For any point P, denote by Ka the Lemoine point of triangle PBC and define Kb, Kc similarly. Triangles ABC and KaKbKc are perspective if and only if P lies on the Lemoine quintic Q016 (adapted from a Hyacinthos message by JeanPierre Ehrmann). Q016 is a bicircular quintic with a real asymptote parallel to the line GK. The tangents at A, B, C pass through K and those at A1, B1, C1 pass through O. The tangents at the isodynamic points X(15), X(16) are those of the Neuberg cubic. Thus, we know the 15 common points of Q016 and the Neuberg cubic. See the analogous Q139. 
