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X(2), X(523), X(2408), X(3413), X(3414)

It is known that any non-pivotal isocubic nK can be considered as the locus of point M such that M and its isoconjugate M* are conjugated with respect to a fixed circle. See Special Isocubics, ยง1.5.3.

This circle is not necessarily unique and one can find nKs such that the fixed circle can be replaced by a pencil of circles. This is the case of K606 = nK0(X523, X69).

The radical axis of the pencil of circles is the Newton line (N) of the quadrilateral formed by the sidelines of ABC and the trilinear polar of the root X(69). (N) is the trilinear polar of X(2996), a line perpendicular at X to the Euler line. X is unlisted in the current edition of ETC. Each circle of the pencil is orthogonal to a fixed circle (C) with center X whose radical axis with the circumcircle is the perpendicular at G to the Euler line. The figure shows this circle (in orange) and two circles (in blue and green) of the pencil.

It follows that, for any point M on K606, the polar line of M in any circle of the pencil passes through the isoconjugate M* of M.

K606 has three real asymptotes : two are the perpendiculars at G to those of the Kiepert hyperbola and the third is perpendicular to the Euler line at the reflection of G in X.

K606 meets the Euler line at G and the two points on (C), the limit-points of the pencil of circles.