X(6), X(111), X(187), X(232), X(248), X(385)
O1, O2 cited in CL051
(Comments below by Wilson Stothers)
R* = X(110) so T(R*) is the Brocard Axis.
K222 contains :
The nodal tangents are the tripolars of the intersections of C(K) and T(R*). These are the points X(1379) and X(1380). Their isoconjugates must be the intersections of T(K) and C(R).
The Lemoine Axis is an asymptote of the pivotal conic as it "meets" it at R. This identifies the conic as it is an inconic of the tangential triangle with center a^2(b^2-c^2)(b^2+c^2-3a^2) : : .
Note that the centre of the pivotal conic then lies on the Lemoine Axis.