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X(357), X(1134), X(1136), X(3272)

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

Let MaMbMc be the Morley triangle. The six trisectors are rotated about each corresponding vertex of ABC of a same angle t, all outwardly or all inwardly. These rotated trisectors define a triangle NaNbNc which is perspective to ABC at P.

When t varies, the locus of P is a nodal cubic we call the Morley - van Tienhoven cubic since Chris van Tienhoven raised the problem in private correspondence. The node is X(3272), the center of the unique inscribed equilateral triangle homothetic to the Morley triangle.

The loci of Na, Nb, Nc are three hyperbolas passing through two vertices of ABC and through the corresponding vertex of the Morley triangle.

See the two figures below.

These points have trilinear coordinates :

Na = sin(B/3+t) sin(C/3+t) : sin(B/3+t) sin(2C/3-t) : sin(C/3+t) sin(2B/3-t), Nb and Nc similarly,

P = sin(A/3+t) / sin(2A/3-t) : sin(B/3+t) / sin(2B/3-t) : sin(C/3+t) / sin(2C/3-t).

Naturally, with t = 0 (mod. π), NaNbNc is the Morley triangle MaMbMc and then P = X(357).

Similarly, with t = 2π/3 (mod. π) and t = -2π/3 (mod. π) we obtain P = X(1134) and P = X(1136) respectively.

With t = π/2 (mod. π), we obtain P = cos(A/3) / cos(2A/3) : : , a point whose isogonal conjugate lies on the line X(356), X(357), X(358).

The vertices of ABC are obtained for t = 2A/3, 2B/3, 2C/3 and the traces on its sidelines for t = -A/3, -B/3, -C/3.

There are two values of t (mod. π) for which P is the node X(3272) of the cubic. The corresponding triangles are represented below.

The isogonal transform of K587 is obtained when the lines passing through a vertex of ABC and the corresponding vertex of the adjunct Morley triangle are rotated in the same way. This (less interesting) cubic contains X(358), X(1135), X1137) and X(3272)*.

When the angles of ABC are tripled, we obtain K589.