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X(1), X(2), X(6), X(43), X(87), X(194), E(566)

excenters

foci of the K-ellipse (inellipse with center K when the triangle ABC is acute angle)

The Grebe cubic is pK(X6, X6) i.e. the isogonal pK with pivot K = X(6), the Lemoine point. See also Z+(L)60.

Locus properties :

1. locus of point P such that the trilinear polars of P and its isogonal conjugate P* are parallel. Thus K102 is a member of the class CL007 of cubics. Now, if we replace trilinear polars by orthotransversals, we obtain the sextic Q021.
2. locus of M such that the circumanticevian triangle of M and the anticevian triangle of its isogonal conjugate M* are perspective (together with another isogonal nK with root G of the class CL020).
3. locus of point P such that the polars of P and P* in the bicevian conic C(P, P*) are parallel.
4. locus of point P such that the center of the bicevian conic C(P, P*) lies on the line PP* (Francisco Javier García Capitán). The locus of this center is K437, a nodal cubic with node K.
5. Kjp = K024 is an isogonal nK0 which meets its asymptotes on a parallel to the trilinear polar of its root. This is true for any nK0(X6, P) with P on the Grebe cubic K102. See also K082.

The three points (other than A, B, C) where K102 meets the circumcircle are the centers of the three isogonal central circular nK0. See isogonal nK0. These three points lie on the rectangular hyperbola passing through X(2), X(3), X(6), X(83), X(99), X(194).

K102 meets the Steiner ellipse again at three points and the tangents at these points are concurrent.