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X(3), X(6), X(3167) = E'

E = isogonal conjugate of X(193) = X(8770)

feet of the symmedians

common points of the Thomson cubic with the circumcircle i.e. vertices of the Thomson triangle (see K346 and a generalization in How pivotal cubics intersect the circumcircle)

Consider a point P(x:y:z) on the circumcircle and its Simson line L. The trilinear pole of L is Q [x/a^2(b^2SB/y - c^2SC/z) : y/b^2(c^2SC/z - a^2SA/x) : z/c^2(a^2SA/x - b^2SB/y)]. The locus of Q is the Simson cubic K010.

The transformation f : P --> Q can be extended to the whole plane although not defined at A, B, C, O. f is now called GS transform in ETC.

Q lies on the line at infinity if and only if P lies on K167 = pK(X184, X6). In other words, f transforms K167 into the line at infinity.

Similarly, f transforms the orthocubic K006 into the trilinear polar of X(264) (isotomic conjugate of O) and the Euler perspector cubic K045 into the line with barycentric equation : a^4 SA x + b^4 SB y + c^4 SC z = 0. See also K168.

The asymptotes of K167 are parallel to those of the isogonal pK whose pivot is the reflection X(3060) of X(51) in G.

The isogonal transform of K167 is K181.

K167 is a member of the pencil generated by the Thomson cubic K002 and the union of the circumcircle and the Brocard axis. This also contains K172, K297.