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K599

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X(4), X(6), X(115), X(187), X(248), X(1989), X(1990)

K599 is the locus of poles of circular pivotal cubics whose orthic line passes through Q = X(140), the midpoint of X(3) and X(5).

When Q = X(2) we obtain K095 and when Q = X(3) we obtain K381.

More generally, when Q lies on the Euler line, we obtain a similar cubic that belongs to the pencil generated by the three cubics above.

This pencil contains a decomposed cubic when Q = X(5) which is the union of the orthic axis and the circumconic through G and K.

All these cubics pass through A, B, C, X(6) (twice), X(1989), X(1990) and two imaginary points on the orthic axis and on the circumconic with perspector X(184).

 

When the pole lies on K599, the pivot lies on a circular cubic that contains X(2), X(4), X(30), X(98), X(265), X(3448) giving K043 = pK(X187, X2), K059 = pK(X1990, X4), K001 = pK(X6, X30), K336 = pK(X248, X98), K060 = pK(X1989, X265) and pK(X115, X3448).