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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 circumellipse. R = (vw)u^2 : (wu)v^2 : (uv)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(vw)(v+w2u) : : , 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 : (vw)u^2 x(w^2y^2+v^2z^2) + cyclic = 0. For example, K218 is the trident with asymptote perpendicular to the Euler line. See also CL030 and a generalization in the page Pconical cubics. 

The real asymptote is the line XW meeting the trilinear polar of R at Z = (vw)^2u^2 : (wu)^2v^2 : (uv)^2w^2 on the inscribed Steiner ellipse. Z is the center of the circumconic 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(vw)x + cyclic = 0. The parabola asymptote has equation : u^2(wu)(uv)[vwx^2  2(u^2  2vw)yz] + cyclic = 0 (obtained with a good help from JeanPierre 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 conicopivotal 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 ยง8. 

See CL063 for another kind of circumtrident. 
