For any point X = u : v : w on the line at infinity, the barycentric square W of X lies on the inscribed Steiner ellipse. R is the intersection of the trilinear polar of X (which is the tangent at W to the inscribed Steiner ellipse) and the polar line of X in the Steiner circum-ellipse. R = (v-w)u^2 : (w-u)v^2 : (u-v)w^2 is a point on K219.
The tripolar centroid TG(X) of X is defined in Clark Kimberling's ETC, preamble before X(1635). For any point X different of G, TG(X) is in fact the isobarycentre of the three points of intersection of the trilinear polar of X with the sidelines of ABC. When X = u : v : w, we have TG(X) = u(v-w)(v+w-2u) : : , and when X lies at infinity, we obtain R as above.
The cubic nK0(W,R) is a trident with point at infinity X which is a flex. Its equation is :
(v-w)u^2 x(w^2y^2+v^2z^2) + cyclic = 0.
The real asymptote is the line XW meeting the trilinear polar of R at Z = (v-w)^2u^2 : (w-u)^2v^2 : (u-v)^2w^2 on the inscribed Steiner ellipse. Z is the center of the circum-conic passing through G and X. The tangent at Z to the inscribed Steiner ellipse passes through R. In other words, the asymptote is the polar line of the root R in the inscribed Steiner ellipse. This asymptote has equation :
vw(v-w)x + cyclic = 0.
The parabola asymptote has equation :
u^2(w-u)(u-v)[vwx^2 - 2(u^2 - 2vw)yz] + cyclic = 0 (obtained with a good help from Jean-Pierre Ehrmann).
The tangent at W to the inscribed Steiner ellipse meets the parallel at G to the real asymptote at Y on the curve.
This trident is obviously a conico-pivotal cubic or cK and the pivotal conic is a parabola tangent to the real asymptote and inscribed in the anticevian triangle of X. See Special Isocubics §8.
See CL063 for another kind of circum-trident.