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X(2), X(51), X(512) 

Let ABC be a given triangle and t a variable parameter. Let P, Q, R be three points on the sidelines BC, CA, AB of triangle ABC such that : BP/PC = CQ/QA = AR/RB = (1  t)/t (with directed lengths) i.e. P is the barycenter of (B,t) and (C,1t), Q and R similarly. K271 is the locus of the circumcenter M of triangle PQR, as the parameter t varies. The centroid of PQR is G for any t. The circumcircle of PQR meets the sidelines again at P', Q', R'. Let G' be the centroid of P'Q'R'. K272 is the locus of G' as t varies. K272 is a nodal cubic with node G passing through X(51), the centroid of the orthic triangle, and X(512) which is a flex at infinity. It has only one real asymptote which is the perpendicular to the Brocard axis (and to the line GX(51)) at the homothetic of X(51) under h(G,1/3). The two other points at infinity are those of the circumellipse through X(670), X(805), X(1576) with perspector the point a^4(b^2+c^2+bc)(b^2+c^2bc): : . The nodal tangents are the lines : (a^4b^2 + a^2c^4  2b^4c^2)x + cyclic = 0 & (a^4c^2 + a^2b^4  2b^2c^4)x + cyclic = 0. The angle of these two tangents is twice the Brocard angle. In other words, each nodal tangent makes an angle with the real asymptote which is the Brocard angle. K272 is an axial cubic invariant under the reflection in the line GX(51). 
