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X(3), X(523), X(2574), X(2575)

antipodes of A, B, C on the circumcircle

All central nK with center O = X(3) have their pole on the trilinear polar of O and their root on the trilinear polar of X(69). They are nK++ and they form the class CL013 of cubics.

The figure above shows K069, the nK with pole X(647), isogonal conjugate of the trilinear pole of the Euler line and also the perspector of the Jerabek hyperbola.

One asymptote is the perpendicular at O to the Euler line and the two others are parallel at O to those of the Jerabek hyperbola.

Its root is X(11064), intersection of the trilinear polars of X(69) and X(99), a point on the line GK, also the barycentric product X(30) x X(69).

See also Special isocubics ยง3.4.1.

K839 is the isogonal transform of K069.