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X(6), X(111), X(187), X(232), X(248), X(385), X(5291), X(9468), X(11081), X(11086), X(17735), X(17961), X(17962), X(17963), X(17964), X(17965), X(17966), X(17967), X(17968), X(17969), X(41368), X(46228)

X(48449) →X(48453)

O1, O2 cited in CL051

See the notations and the general properties at CL030. See here for a family of related cubics.

A line passing through G meets K018 again at two points P1, P2 which lie on a circum-conic passing through K. The barycentric product P1 x P2 lies on K222.

(Comments below by Wilson Stothers)

K222 is

1. the locus of X such that XX* is divided harmonically by the Brocard (T(R*)) and Lemoine Axes (T(F)),
2. the K-Hirst inverse of C(R) - the circumconic through G and K. C(R) contains many named points including X(25) and X(1989).
3. the isogonal transform of K185.

R* = X(110) so T(R*) is the Brocard Axis.

K222 contains :

• K, the node,
• X(111) the intersection of C(K) and C(R),
• X(187) as the intersection of T(R*) and T(K),
• X(232) as the K-Hirst inverse of X(25),
• X(385) as the K-Hirst inverse of G.

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 X(8651) = 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.