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X(2), X(4), X(6), X(25), X(193), X(371), X(372), X(2362) foci of the Kellipse (inellipse with center K when the triangle ABC is acute angle) isogonal conjugates of the CPCC or Hcevian points, see Table 11 

The trilinear polar of P meets the sidelines of ABC at U, V, W. Ub, Uc are the projections of U on AC, AB and Vc, Va, Wa, Wb are defined similarly. These triangles UbVcWa and UcVaWb are orthologic if and only if P lies on K233 or on the cubic K232 which is in fact the locus of P such that Uc, Va, Wb are collinear. K233 is a pivotal cubic with asymptotes parallel to those of pK(X6,X69) = K169 and pK(X2, X315). It is the isogonal transform of pK(X3, X2) = K168 and the isotomic transform of pK(X305, X76). It meets the circumcircle at the same points as pK(X6, X193). This cubic is a member of the class CL042 and also a member of the class CL043 : it meets the circumcircle at A, B, C and three other points where the tangents are concurrent. 
