X(1), X(4), X(56), X(145), X(218), X(279), X(1433)
vertices A', B', C' of the intouch triangle (pedal triangle of I)
projections of I on the altitudes
infinite points of K692
points of K1086 = pK(X6, X145) on the circumcircle
K360 is the Lemoine generalized cubic K(X1) hence it is a nodal cubic with node I, the nodal tangents being parallel to the asymptotes of the Feuerbach hyperbola.
The tangents at A, B, C are the symmedians.
The common points Q1, Q2, Q3 with C(O,R) are those of K1086 = pK(X6, X145).
Note that ABC and Q1Q2Q3 share the same incircle and the tangents at Q1, Q2, Q are concurrent. Hence K360 is also a psK in Q1Q2Q3. These latter properties remain true for any cubic psK(X56, X7, ?) and, in particular, K631 = pK(X56, X7).
The tangents to K360 at Q1, Q2, Q3 concur at X = X(15306) = X(1)X(227) /\ X(56)X(902) /\ X(109)X(999).
K360 is the isogonal transform of K259 = spK(X145, X1).
K577 is the X(56)-isoconjugate of K360.
See the related K1058 = psK(X1254, X7, X1).
K259 = spK(X145, X1) and K360 = spK(X8, X1) generate a pencil of nodal circum-cubics stable under isogonal conjugation. The only self-isogonal cubics are spK(X1, X1) which is the union of the internal bisectors of ABC and K086 = spK(X519, X1), the Gergonne strophoid. K915 = spK(X10, X1) is another example.
Any cubic of the pencil is spK(P, X1) for some point P on the line X(1)X(2) and its isogonal transform is spK(P', X1) where P' is the reflection of P about X(1).
spK(P, X1) meets C(O,R) again at P1, P2, P3 and the sidelines of P1P2P3 are tangents to the parabola with focus X(106) and directrix the line X(1)X(2). Note that these are the singular focus and orthic line of K086 respectively.