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A, B, C singular points points at infinity of the sidelines of ABC vertices Ga, Gb, Gc of the antimedial triangle other points below 

Q006 is a circular quintic with singular focus X(5). The three real asymptotes are the sidelines of the medial triangle. The tangents at Ga, Gb, Gc are parallel to the sidelines of the tangential triangle. Locus properties




When ABC is obtusangle, Q006 meets the de Longchamps axis (trilinear polar of X(76), isotomic transform of the circumcircle) at two real points P1, P2 lying on the circumcircle and also on the circumcircle of GaGbGc. P1 and P2 are clearly isotomic conjugates with midpoint X(858), the projection of X(3) on the de Longchamps axis, also the inverse of X(2) in the nine point circle. The remaining three common points P3, P4, P5 lie on the nonpivotal cubic nK0(X2, Q) where Q is the intersection of the lines X(2)X(39) and X(5)X(83). Q = a^4 + b^4 + c^4  b^2c^2 : : is now X(7828) in ETC. 

The cevian triangles of P1 and P2 are congruent to the cevian triangles of X(110) and X(850). These points lie on the cubic nK(X2, X2, X110), a Tucker cubic. More details on Tucker cubics here. 



Q006 and the X99Cevian points Q006 contains the X99Cevian points. See Table 28 : cevian and anticevian points for explanations. 

There are seven X99Cevian points Pi although not always all real. These also lie on the Steiner ellipse of the antimedial triangle. If A'B'C' is the cevian triangle of one of these points Pi then Pi is the Steiner point of the triangle A'B'C'. The figure shows a configuration where three of them are real. (C) and (S) are the circumcircle and the Steiner ellipse of the cevian triangle of P1. G1 is the centroid of the triangle. 
