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X(2), X(63), X(394), X(493), X(494), X(3083), X(3084), X(5408), X(5409), X(6503), X(6504), X(6505), X(6513), X(8222), X(8223)

isotomic conjugates of X(1585), X(1586), now X(11090) and X(11091) in ETC (2016-12-02)

midpoints of ABC with tangents passing through X(6503)

vertices of the anticevian triangle of X(63)

foci of the inconic with center X(3757)

K857 is the isogonal transform of K678, the isotomic transform of pK(X2052, X2052) see CL007, the complement of pK(X2, X6515).

K857 is also :

• the barycentric quotient of K006 by X(4). Recall that the barycentric product of K006 by X(4) is K678. K857 is therefore anharmonically equivalent to the Orthocubic K006.

• the transform of the Orthocubic K006 under the mapping f described in the page K856 which is an analogous cubic.

• the transform of the Thomson cubic K002 under the symbolic substitution SS{a –> cosA}. See ETC, X(6503) and below.

K857 and K678 generate a pencil which contains nK0(X6, X512) and pK(X6, X3757). It follows that all these cubics pass through the foci of the inconic with center X(3757).