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K867

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on both curves :

X(2), X(298), X(299), X(1494),

infinite points of the Steiner ellipses

on K867a = pK(X298, X299)

X(13), X(533), X(617), X(11078), X(11129)

isotomic conjugates of the 3 (always real) mates of X(370), see Table 10

on K867b = pK(X299, X298)

X(14), X(532), X(616), X(11092), X(11128)

isotomic conjugates of X(370) and the 2 (not always real) mates of X(370), see Table 10

K867a and K867b are related with CL041. They belong to the grey group in the large table of cubics which also contains K419a and K419b, K859a and K859b.

Each is the X(2)-Hirst inverse of the other since X(298), X(299) are themselves X(2)-Hirst inverse points. Note that X(11078), X(11092) are also the X(2)-Hirst inverses of X(13), X(14) respectively.

Their isotomic transforms are K419a and K419b respectively.

K419a and K419b are the barycentric products of K867a by X(13), K867b by X(14) respectively.

K867a and K867b generate a pencil of circum-cubics which contains pK(X1494, X1494) and the decomposed cubic into the Steiner ellipse and the line passing through X2, X6, X69, X81, X86, X141, X183, X193, X230, X298, X299, etc.