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K718

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X(2), X(4), X(114), X(193), X(264), X(287), X(297), X(325), X(511), X(1916)

infinite points of the Steiner ellipses

S = isotomic conjugate of X(3564)

vertices of the orthic triangle

K718 is the only pK with pivot H passing through the infinite points of the Steiner ellipses. See the related CL019.

It is the isogonal pK with pivot X(325) with respect to the orthic triangle or, equivalently, the locus of M such that X(325), M and H/M are collinear (H/M is the H-Ceva conjugate of M).

See a generalization in CL041.

K718 is the cornerstone of a group of 12 cubics all related between themselves under isogonal, isotomic, G-Hirst conjugations – denoted g, t, h in the following diagram – or a product of these such as e = gtg which is X(32)-isoconjugation. All these cubics are strong cubics and contain a good number of ETC centers.

A similar group of weak cubics is obtained when K718 is replaced with K323.

K323hexa

K781

K323hexa

K786

K323hexa
K323hexa
K323hexa

K718

K323hexa

K780

K323hexa
K323hexa
K323hexa

K785

K782

K323hexa

K776

K323hexa

K779

K323hexa
K323hexa
K323hexa
K323hexa

K777

K323hexa

K778

K323hexa
K323hexa
K323hexa

K783

K323hexa

K784

K323hexa
K323hexa