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See also K584 for a trilinear equation and comments and the related cubics K585 and K586.

X(1), X(356), X(357), X(358), X(1134), X(1135), X(3605) = X(356)* = isogonal conjugate of X(356)

excenters Ia, Ib, Ic, cevians of X(356)

vertices A1, B1, C1 of Morley (first) triangle (green)

vertices A1*, B1*, C1* of adjunct Morley triangle (yellow)

P = X(1507) perspector of triangles A1B1C1 and IaIbIc, its isogonal conjugate P*

Q= X(1508) perspector of triangles A1*B1*C1* and IaIbIc, its isogonal conjugate Q*

traces of X(358) on the Morley sidelines

The Morley (first) cubic is the isogonal pK with pivot X(356) center of the Morley (first) triangle. The two perspectors seen above have barycentrics :

P = [a(1 + 2( - cosA/3 + cosB/3 + cosC/3)) : : ] on the line X(1)X(358)

Q = [a(2 - secA/3 + secB/3 + secC/3) : : ] on the line X(1)X(357)

Note that :

  • X(356) is the perspector of A1B1C1 and A1*B1*C1*,
  • X(357) is the perspector of ABC and A1B1C1,
  • X(358) is the perspector of ABC and A1*B1*C1*.

See also table 9 : Morley & Co.

We find this cubic in a paper by J. Neuberg published in Mathesis, vol. 37, pp.356-367 (1923) under the title "Sur les trisectrices des angles d'un triangle". Neuberg also examines the cubic K030 but surprisingly not the cubic K031. He also generalizes to another cubic related to one of the 18 equilateral Morley triangles.

There are indeed 18 Morley cubics among them 3 "central" cubics namely K029, K030, K031. These latter cubics are those obtained with the trisectrices of the same angles in ABC i.e. the angles A+2 k1 π, B + 2 k2 π, C + 2 k3 π where k1 = k2 = k3.

The figure below shows the cubic obtained with k1 = 1, k2 = k3 = -1. The points denoted Ea, Va, Wa are the equivalents of X(356), X(357), X(358). XYZ and X*Y*Z* are the corresponding Morley and adjunct Morley triangles.