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X(2), X(3), X(6), X(69), X(485), X(486), X(8770)

X(2)-Ceva conjugate of X(69)


CPCC or H-cevian points, see Table 11

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

K168 is the pivotal cubic whose GST (see here for a definition) is the de Longchamps line. See K167. See also K007, property 8 and K002, property 12.

It is also the locus of point P such that P, GSC(P), X(6) are collinear. GSC is defined here.

It has the same asymptotic directions as pK(X6, X193) and K170 = pK(X2, X4) which is actually its anticomplement.

It meets the circumcircle at the same points as pK(X6, X69) = K169 and the Steiner circum-ellipse at the same points as pK(X2, X76) = K141. It is tangent at O to the Brocard line, at G to the Euler line.

Its isogonal transform is pK(X25, X4) = K233 and its isotomic transform is pK(X264, X264). See CL007.

It is therefore anharmonically equivalent to the Orthocubic K006.

It is the image of the cubic K345 under the symbolic substitution SS{a -> SA}, also the image of the cubic K1077 under the symbolic substitution SS{a -> a^2}

K168 is an isogonal pK with pivot X(6) with respect to the complement of the orthic triangle.