Morley's trisector theorem

In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles.

There are many proofs of Morley's theorem, some of which are very technical. Several early proofs were based on delicate trigonometric calculations. The first published geometric proof was given by M. T. Naraniengar in 1909. Recent proofs include an algebraic proof by Alain Connes (1998, 2004) extending the theorem to general fields, and John Conway's elementary geometry proof. The latter starts with an equilateral triangle and shows that a triangle may be built around it which will be similar to any selected triangle. Morley's theorem does not hold in spherical and hyperbolic geometry.

One proof uses the trigonometric identity

Points D,E,F are constructed on BC as shown. Clearly α+β+γ = 60° therefore ∠CYA = 120°+β and the angles of ΔXEF are α, 60°+β, 60°+γ. Now sin(60°+β) = DX/XE and AC/sin(120°+β) = AY/sin γ by the sine rule, so the height h of ΔABC is given by

As the numerators are equal, XE.AY = XF.AZ. But ∠EXF = ∠ZAY and the sides about these angles are in the same ratio (because XE/XF = AZ/AY) so the triangles XEF and AZY must be similar. Thus the base angles of ΔAZY are 60°+β and 60°+γ. Similar arguments yield the base angles of ΔBXZ and ΔCYX and all the angles in the figure can now be easily determined.

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