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X(2), X(4), X(6), X(25), X(193), X(371), X(372), X(2362), X(7133), X(14248), X(16232), X(20034)

vertices of the orthic triangle

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

isogonal conjugates of the CPCC or H-cevian 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 the isogonal transform of pK(X3, X2) = K168 and the isotomic transform of pK(X305, X76). It is therefore anharmonically equivalent to the Orthocubic K006.

K233 is the image of the cubic K366 under the symbolic substitution SS{a -> SA}.

The symbolic substitution SS{a -> a^2} maps K365 onto K233.

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 Q1, Q2, Q3 where the tangents are concurrent.

Since the isopivot of K233 is X(6), the polar conic of X(6) is a circum-conic, namely the Jerabek hyperbola (J).

K233 is a pivotal cubic with asymptotes parallel to those of pK(X6, X69) = K169 and pK(X2, X315).

More generally, for Ω on K177 = pK(X32, X2) and P on pK(X2, X315), one can find a pK(Ω, P) with the same infinite points as K233.

K233 meets the circumcircle (O) at the same points Q1, Q2, Q3 as pK(X6, X193). The orthic inconic (C) is also inscribed in the triangle Q1Q2Q3 with contacts R1, R2, R3 with its sidelines. These latter points lie on K233.

This triangle Q1Q2Q3 is then a poristic triangle with respect to (O) and (C).

More generally, for Ω on pK(X32 x X25, X25) and P on K233, one can find a pK(Ω, P) with the same points on the circumcircle as K233.

Note that pK(X32 x X25, X25) is the barycentric product X(25) x K168.

The tangents at Q1, Q2, Q3 to K233 concur at X which is not on K233. It follows that K233 is a psK in the triangle Q1Q2Q3.

X = a^2 (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) (a^4 - 4 a^2 b^2 + 3 b^4 - 4 a^2 c^2 - 26 b^2 c^2 + 3 c^4) : : , SEARCH = -1.87280606149532.

X lies on the line X(6), X(3531). It is the barycentric product X(4) x X(5544).

***

K1313 = pK(X14575, X6) is the locus of the pseudo-poles of central cubics psK with center X(6).

The loci of the pseudo-pivots and the pseudo-isopivots are K233 and K1047 respectively.

See Pseudo-Pivotal Cubics and Poristic Triangles and Central cubics.