K082 is the locus of point P such that the power of P with respect to its pedal circle is k = - S tan w, where S = area (ABC) and w = Brocard angle.
The cubic is an isogonal nK0+ with root X(2) = G, a member of the class CL020 of cubics.
K082 is the only case of such cubic with three concurring asymptotes (at K = X(6)). These meet the cubic again at three points on the trilinear polar of X(251) and on the tangents at A, B, C to the circumcircle. The isogonal conjugates of these three points lie on the circum-conic with center X(39), on the sidelines of the antimedial triangle, on the cevian lines of X(194) in this latter triangle.
The equation of K082 is remarkably simple.
The centers of the polar conics of A, B, C are the vertices of the third Brocard triangle.
K082 is the locus of point M such that M and its isogonal conjugate M* are conjugated with respect to :
– the Steiner ellipse,
– the circle (C) with center H that contains the intersections of the circumcircle and the perpendicular at K to the Euler line,
– any conic of the pencil generated by the ellipse and the circle above.
This pencil contains the diagonal rectangular hyperbola (H) with center X(98) that contains the in/excenters.
See a generalization in Table 4.