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X(2), X(3), X(5), X(13), X(14), X(110), X(399), X(3448), X(6592)

X(39162), X(39163), X(39164), X(39165) : foci of the Steiner inellipse

points on (O) and pK(X6, X37779)

other points below

Geometric properties :

K1296 is a focal cubic that passses through the foci of the Steiner inellipse. See Table 48 for other examples.

It is the locus of point M such that X(399), M, Psi(M) are collinear.

The singular focus is F = Psi(X399), on the line {476,549}, on the Hutson-Parry circle (Psi-image of the Fermat axis), on the cubic K883. F also lies on every circle that is the Psi-image of a line passing through X(399), for instance the circles {2, 3, 110, 842} and {2, 125, 140, 1116}.

K1296 meets the line at infinity at Z on the lines {2,399}, {110,549}, {125,547}, {428,1986}, etc.

K1296 is also the locus of foci of conics with center on the line {2,399} which are inscribed in triangles {2, 3, 110}, {2, 13, 14} and, more generally, in every triangle {G, M, Psi(M)} where M is a point on the curve.

As such, it is an isogonal nK with respect to each triangle. In particular, Y(13) and Y(14) are the isogonal conjugates of X(13) and X(14) with respect to triangle {2, 3, 110}. These two points are collinear with X(399) hence they are Psi-homologous.

Z and F are also isogonal conjugate points on K1296 which must contain the isogonal conjugate of X(6592).

K1296 and K067 have six common points on the (green) Lester circle. They meet again at X(6592) and two imaginary points on the line {6592,10272} and on the circum-conic passing through {110,1173}. They generate a pencil of circular cubics that contains K1297, the complement of the Neuberg cubic K001.