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X(2), X(6), X(53), X(61), X(62), X(251)

Ka, Kb, Kc :vertices of the cevian triangle of X(6)

foci of the Steiner in-ellipse

Sa, Sb, Sc : see below

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 Jean-Pierre Ehrmann). The locus of the perspector is the Lemoine sextic Q156.

A, B, C are nodes on Q156 with nodal tangents the symedians and the tangents to K755. Q156 meets the sidelines of ABC again at Ka, Kb, Kc and Sa, Sb, Sc also on K755.

Sa is the trace on BC of the line which is the reflection in K of the line A, X(141). Sb and Sc are defined likewise. Recall that K755 is spK(X141, X6) as in CL055.

Q156 meets the Brocard axis at X(61), X(62) and K counted four times.