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X(3), X(67), X(98), X(468), X(1691), X(3479), X(3480), X(3564), X(5061), X(6177), X(6178), X(13414), X(13415), X(34138), X(34310), X(41175)

X(51454) → X(51459)

Geometric properties :

Every non-isogonal focal nK passing through X(99) has its pole Ω on the trilinear polar of X(99), its root R on the trilinear polar of X(670) and its singular focus F on K1281, the isogonal transform of K289.

For instance, with Ω = G, the cubic is K091 and R = X(11056), F = X(67).

K1281 is a circular nodal cubic with node O and singular focus the midpoint X(51460) of X(3), X(2079), also the inverse in (O) of X(6321).

K1281 is the inverse in (O) of the circum-conic (C) with perspector X(3049) hence the nodal tangents are parallel to the asymptotes of (C).

Note that (C) is the isogonal transform of the line (L) passing through H and X(69). The antigonal transform of (L) is the nodal cubic K289.

(L) meets the cevian lines of X(512) in the antimedial triangle GaGbGc at three points. The inverses in (O) of the isogonal conjugates of these points are the points of K1281 on the sidelines of ABC.


On the other hand, the isogonal focal nKs passing through X(99) form a pencil and the singular focus is always X(99). The root also lies on the trilinear polar of X(670). Obviously, these cubics pass through X(512).

See K084, the Steiner isogonal central focal cubic, nK(X6, X11052, X99), spK(X512, X99).