X(1), X(3), X(8), X(220), X(277), X(3160)
A', B', C' : midpoints of ABC
points at infinity of pK(X6, X145)
vertices of the Nagel triangle, see K692
A1, B1, C1 : pedals of X(1) in the medial triangle
X1-OAP points, see also Table 53
K259 is a member of the class CL033 (Deléham cubics).
The nodal tangents at X(1) are parallel to the asymptotes of the Feuerbach hyperbola. The tangents at A, B, C concur at X(55).
For any point Q on the line X(2)X(7), the trilinear polar of Q meets the lines IA', IB', IC' at Qa, Qb, Qc. ABC and QaQbQc are perspective and the perspector is a point on K259. This gives a simple way for finding a lot of reasonably simple points on the curve.
See the related sextic Q104.
K259 is actually the generalized Lemoine cubic with node X(1) with respect to the medial triangle.
The isogonal transform of K259 is K360.
The anticomplement of K259 is the generalized Lemoine cubic with node X(8).