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K913

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X(1), X(2), X(3), X(30), X(110), X(147), X(399), X(5536)

excenters

vertices of the Thomson triangle

K913 is a member of the pencil generated by two decomposed cubics : one is the union of the Euler line and the circumcircle, the other is the union of the line at infinity and the Thomson-Jerabek hyperbola. This pencil is stable under isogonal conjugation with respect to the Thomson triangle. It also contains K463, K834 which are the only self-isogonal cubics.

All these circular cubics have their focus on the line X(3), X(74), X(110), etc. That of K913 is F, the homothetic of X(110) under h(X3, 3). F is now X(14094) in ETC (2017-08-07).

Note that the foci of a cubic and its isogonal transform are inverse in the circumcircle of ABC.

This pencil is actually the pencil K001, K187, etc, for the Thomson triangle. See K187 for further details.