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See below

vertices of the Morley triangle

vertices of the adjunct Morley triangle

A', B', C' traces of the trilinear polar of the root X(3602), the harmonic conjugate of X(356) wrt X(357) and X(358). These points lie on the sidelines of the two triangles above


K029 and K584 are two isogonal cubics passing through the vertices of the Morley triangle and the adjunct Morley triangle (see TCCT p. 124 and p. 165)

K029 is a pK with pivot the center X(356) of the Morley triangle.

K584 is a nK with root R defined above.

There is an interesting similarity between the isogonal cubics K005 (Napoleon cubic), K397 = nK(X6, X30, X2) and K029 (first Morley cubic), K584 (Morley isogonal nK) when we compare their trilinear equations.


We see that the cubics K029 and K584 are obtained from K005 and K397 when the angles of ABC are trisected.

Note that this same operation maps the points X(5), X(4), X(3) to the Morley points X(356), X(357), X(358) and X(30) to the root X(3602) of K584.

K005 and K397 meet at A, B, C, the projections of X(2) on the altitudes, their isogonal conjugates. The same mapping transforms these six latter points into the six vertices of the two Morley triangles.


With this same technique, we transform the Neuberg cubic K001 into K585 with trilinear equations


and the Kn cubic K060 into K586 with trilinear equations