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X(2), X(99), X(523) 

This is the only circular isotomic cK with node at G. See Special isocubics §8.5.2. Its root is X(11054), on the line X(2)X(39).The singular focus is X(14653). 

K088 is the Psiimage of the conic passing through G and the vertices of the second Brocard triangle. The tangent at G is parallel to the lines 74,98 99,110 113,114 115,125. Psi is the involution that is the product of a symmetry about one axis of the Steiner inellipse and the inversion in the circle with diameter F1F2 (the foci of the ellipse). More details in "Orthocorrespondence and Orthopivotal Cubics", §5. 

K088 is a member of the pencil of isotomic nKs generated by the two following decomposed cubics : • union of the Steiner ellipse and the line at infinity, • union of the circumcircle and the de Longchamps line, the trilinear polar of X(76) and the isotomic transform of the circumcircle. Each cubic is circular with singular focus on the line X(3), X(67), etc. The root lies on the line X(2), X(39), etc. Special cubics of the pencil : • K088 which is a cK with node X(2) and three other cKs with nodes the vertices of the antimedial triangle. • K091 which is a focal cubic. • K197 which is the only nK0 of the pencil. • an axial cubic (K) with


(K) meets its axis of symmetry at X(99) and two other points M1, M2 which are isotomic conjugates hence lying on the conic (C), the isotomic transform of its axis. (C) is a circumhyperbola passing through X(67), X(98), X(523), X(542), X(1494), X(1989). Obviously, the tangents at X(99), M1, M2 are parallel to the asymptote hence perpendicular to the Euler line. 
