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See K584

X(356), X(357), X(3602), X(3604), X(5456)

X(3274)* = isogonal conjugate of X(3274) = X(356)' = harmonic conjugate of X(356) wrt X(357) and X(358)

P = X(5456) = sin 2A/3 : sin 2B/3 : sin 2C/3 is the pivot

Y(79) and Y(80) described and represented below

K586 is a pivotal cubic obtained from the Kn cubic K060 when the angles of ABC are trisected. See K584 for further details.

Its pole is X(3275) = a sin A/3 : b sin B/3 : c sin C/3 and its pivot is X(5456) as above. The isopivot is X(357).

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K586 is the locus of point M whose cevian triangle is perspective with the Morley triangle. The locus of the perspector is the first Morley cubic K029.

When cevian triangle is replaced with anticevian triangle, we obtain K585.

For any point M on K586, the cevian triangle A'B'C' of M is perspective to the Morley triangle and the perspector N lies on the first Morley cubic K029.

The figure represents M = X(3602) = X(3274)* and N = X(358).

Y79 and Y80 are two points obtained from X(79) and X(80), these two latter points lying on the cubic K060 from which K586 derives.

In trilinear coordinates, we have :

X(79) = 1 / (1 + 2 cos A),

X(80) = 1 / (1 - 2 cos A),

thus

Y(79) = 1 / (1 + 2 cos A/3),

Y(80) = 1 / (1 - 2 cos A/3).

Note that these two points are isoconjugate hence collinear with the pivot P.

This is also true for the points X(3273)* and X(3274)* obtained from the Fermat points X(13) and X(14).