Morley's trisector theorem

There are many proofs of Morley's theorem, some of which are very technical.[1] Several early proofs were based on delicate trigonometric calculations. Recent proofs include an algebraic proof by Alain Connes (1998, 2004) extending the theorem to general fields other than characteristic three, and John Conway's elementary geometry proof.[2][3] 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[4] and hyperbolic geometry.

Fig 1.   Elementary proof of Morley's trisector theorem

One proof uses the trigonometric identity

${\displaystyle \sin(3\theta )=4\sin \theta \sin(60^{\circ }+\theta )\sin(120^{\circ }+\theta )}$

(1)

which, by using of the sum of two angles identity, can be shown to be equal to

${\displaystyle \sin(3\theta )=-4\sin ^{3}\theta +3\sin \theta .}$

The last equation can be verified by applying the sum of two angles identity to the left side twice and eliminating the cosine.

Points ${\displaystyle D,E,F}$ are constructed on ${\displaystyle {\overline {BC}}}$ as shown. We have ${\displaystyle 3\alpha +3\beta +3\gamma =180^{\circ }}$, the sum of any triangle's angles, so ${\displaystyle \alpha +\beta +\gamma =60^{\circ }.}$ Therefore, the angles of triangle ${\displaystyle XEF}$ are ${\displaystyle \alpha ,(60^{\circ }+\beta ),}$ and ${\displaystyle (60^{\circ }+\gamma ).}$

From the figure

${\displaystyle \sin(60^{\circ }+\beta )={\frac {\overline {DX}}{\overline {XE}}}}$

(2)

and

${\displaystyle \sin(60^{\circ }+\gamma )={\frac {\overline {DX}}{\overline {XF}}}.}$

(3)

Also from the figure

${\displaystyle \angle {AYC}=180^{\circ }-\alpha -\gamma =120^{\circ }+\beta }$

and

${\displaystyle \angle {AZB}=120^{\circ }+\gamma .}$

(4)

The law of sines applied to triangles ${\displaystyle AYC}$ and ${\displaystyle AZB}$ yields

${\displaystyle \sin(120^{\circ }+\beta )={\frac {\overline {AC}}{\overline {AY}}}\sin \gamma }$

(5)

and

${\displaystyle \sin(120^{\circ }+\gamma )={\frac {\overline {AB}}{\overline {AZ}}}\sin \beta .}$

(6)

Express the height of triangle ${\displaystyle ABC}$ in two ways

${\displaystyle h={\overline {AB}}\sin(3\beta )={\overline {AB}}\cdot 4\sin \beta \sin(60^{\circ }+\beta )\sin(120^{\circ }+\beta )}$

and

${\displaystyle h={\overline {AC}}\sin(3\gamma )={\overline {AC}}\cdot 4\sin \gamma \sin(60^{\circ }+\gamma )\sin(120^{\circ }+\gamma ).}$

where equation (1) was used to replace ${\displaystyle \sin(3\beta )}$ and ${\displaystyle \sin(3\gamma )}$ in these two equations. Substituting equations (2) and (5) in the ${\displaystyle \beta }$ equation and equations (3) and (6) in the ${\displaystyle \gamma }$ equation gives

${\displaystyle h=4{\overline {AB}}\sin \beta \cdot {\frac {\overline {DX}}{\overline {XE}}}\cdot {\frac {\overline {AC}}{\overline {AY}}}\sin \gamma }$

and

${\displaystyle h=4{\overline {AC}}\sin \gamma \cdot {\frac {\overline {DX}}{\overline {XF}}}\cdot {\frac {\overline {AB}}{\overline {AZ}}}\sin \beta }$

Since the numerators are equal

${\displaystyle {\overline {XE}}\cdot {\overline {AY}}={\overline {XF}}\cdot {\overline {AZ}}}$

or

${\displaystyle {\frac {\overline {XE}}{\overline {XF}}}={\frac {\overline {AZ}}{\overline {AY}}}.}$

Since angle ${\displaystyle EXF}$ and angle ${\displaystyle ZAY}$ are equal and the sides forming these angles are in the same ratio, triangles ${\displaystyle XEF}$ and ${\displaystyle AZY}$ are similar.

Similar angles ${\displaystyle AYZ}$ and ${\displaystyle XFE}$ equal ${\displaystyle (60^{\circ }+\gamma )}$, and similar angles ${\displaystyle AZY}$ and ${\displaystyle XEF}$ equal ${\displaystyle (60^{\circ }+\beta ).}$ Similar arguments yield the base angles of triangles ${\displaystyle BXZ}$ and ${\displaystyle CYX.}$

In particular angle ${\displaystyle BZX}$ is found to be ${\displaystyle (60^{\circ }+\alpha )}$ and from the figure we see that

${\displaystyle \angle {AZY}+\angle {AZB}+\angle {BZX}+\angle {XZY}=360^{\circ }.}$

Substituting yields

${\displaystyle (60^{\circ }+\beta )+(120^{\circ }+\gamma )+(60^{\circ }+\alpha )+\angle {XZY}=360^{\circ }}$

where equation (4) was used for angle ${\displaystyle AZB}$ and therefore

${\displaystyle \angle {XZY}=60^{\circ }.}$

Similarly the other angles of triangle ${\displaystyle XYZ}$ are found to be ${\displaystyle 60^{\circ }.}$

The first Morley triangle has side lengths[5]

$${\displaystyle a^{\prime }=b^{\prime }=c^{\prime }=8R\,\sin {\tfrac {1}{3}}A\,\sin {\tfrac {1}{3}}B\,\sin {\tfrac {1}{3}}C,}$$

where R is the circumradius of the original triangle and A, B, and C are the angles of the original triangle. Since the area of an equilateral triangle is ${\displaystyle {\tfrac {\sqrt {3}}{4}}a'^{2},}$ the area of Morley's triangle can be expressed as

$${\displaystyle {\text{Area}}=16{\sqrt {3}}R^{2}\,\sin ^{2}\!{\tfrac {1}{3}}A\,\sin ^{2}\!{\tfrac {1}{3}}B\,\sin ^{2}\!{\tfrac {1}{3}}C.}$$

Morley's theorem entails 18 equilateral triangles. The triangle described in the trisector theorem above, called the first Morley triangle, has vertices given in trilinear coordinates relative to a triangle ABC as follows:

$${\displaystyle {\begin{array}{ccccccc}A{\text{-vertex}}&=&1&:&2\cos {\tfrac {1}{3}}C&:&2\cos {\tfrac {1}{3}}B\\[5mu]B{\text{-vertex}}&=&2\cos {\tfrac {1}{3}}C&:&1&:&2\cos {\tfrac {1}{3}}A\\[5mu]C{\text{-vertex}}&=&2\cos {\tfrac {1}{3}}B&:&2\cos {\tfrac {1}{3}}A&:&1\end{array}}}$$

Another of Morley's equilateral triangles that is also a central triangle is called the second Morley triangle and is given by these vertices:

$${\displaystyle {\begin{array}{ccccccc}A{\text{-vertex}}&=&1&:&2\cos {\tfrac {1}{3}}(C-2\pi )&:&2\cos {\tfrac {1}{3}}(B-2\pi )\\[5mu]B{\text{-vertex}}&=&2\cos {\tfrac {1}{3}}(C-2\pi )&:&1&:&2\cos {\tfrac {1}{3}}(A-2\pi )\\[5mu]C{\text{-vertex}}&=&2\cos {\tfrac {1}{3}}(B-2\pi )&:&2\cos {\tfrac {1}{3}}(A-2\pi )&:&1\end{array}}}$$

The third of Morley's 18 equilateral triangles that is also a central triangle is called the third Morley triangle and is given by these vertices:

$${\displaystyle {\begin{array}{ccccccc}A{\text{-vertex}}&=&1&:&2\cos {\tfrac {1}{3}}(C+2\pi )&:&2\cos {\tfrac {1}{3}}(B+2\pi )\\[5mu]B{\text{-vertex}}&=&2\cos {\tfrac {1}{3}}(C+2\pi )&:&1&:&2\cos {\tfrac {1}{3}}(A+2\pi )\\[5mu]C{\text{-vertex}}&=&2\cos {\tfrac {1}{3}}(B+2\pi )&:&2\cos {\tfrac {1}{3}}(A+2\pi )&:&1\end{array}}}$$

The first, second, and third Morley triangles are pairwise homothetic. Another homothetic triangle is formed by the three points X on the circumcircle of triangle ABC at which the line XX ⁻¹ is tangent to the circumcircle, where X ⁻¹ denotes the isogonal conjugate of X. This equilateral triangle, called the circumtangential triangle, has these vertices:

$${\displaystyle {\begin{array}{lllllll}A{\text{-vertex}}&=&{\phantom {-}}\csc {\tfrac {1}{3}}(C-B)&:&{\phantom {-}}\csc {\tfrac {1}{3}}(2C+B)&:&-\csc {\tfrac {1}{3}}(C+2B)\\[5mu]B{\text{-vertex}}&=&-\csc {\tfrac {1}{3}}(A+2C)&:&{\phantom {-}}\csc {\tfrac {1}{3}}(A-C)&:&{\phantom {-}}\csc {\tfrac {1}{3}}(2A+C)\\[5mu]C{\text{-vertex}}&=&{\phantom {-}}\csc {\tfrac {1}{3}}(2B+A)&:&-\csc {\tfrac {1}{3}}(B+2A)&:&{\phantom {-}}\csc {\tfrac {1}{3}}(B-A)\end{array}}}$$

A fifth equilateral triangle, also homothetic to the others, is obtained by rotating the circumtangential triangle π/6 about its center. Called the circumnormal triangle, its vertices are as follows:

$${\displaystyle {\begin{array}{lllllll}A{\text{-vertex}}&=&{\phantom {-}}\sec {\tfrac {1}{3}}(C-B)&:&-\sec {\tfrac {1}{3}}(2C+B)&:&-\sec {\tfrac {1}{3}}(C+2B)\\[5mu]B{\text{-vertex}}&=&-\sec {\tfrac {1}{3}}(A+2C)&:&{\phantom {-}}\sec {\tfrac {1}{3}}(A-C)&:&-\sec {\tfrac {1}{3}}(2A+C)\\[5mu]C{\text{-vertex}}&=&-\sec {\tfrac {1}{3}}(2B+A)&:&-\sec {\tfrac {1}{3}}(B+2A)&:&{\phantom {-}}\sec {\tfrac {1}{3}}(B-A)\end{array}}}$$

An operation called "extraversion" can be used to obtain one of the 18 Morley triangles from another. Each triangle can be extraverted in three different ways; the 18 Morley triangles and 27 extravert pairs of triangles form the 18 vertices and 27 edges of the Pappus graph.[6]

The Morley center, X(356), centroid of the first Morley triangle, is given in trilinear coordinates by

$${\displaystyle \cos {\tfrac {1}{3}}A+2\cos {\tfrac {1}{3}}B\,\cos {\tfrac {1}{3}}C\,:\,\cos {\tfrac {1}{3}}B+2\cos {\tfrac {1}{3}}C\,\cos {\tfrac {1}{3}}A\,:\,\cos {\tfrac {1}{3}}C+2\cos {\tfrac {1}{3}}A\,\cos {\tfrac {1}{3}}B}$$

1st Morley–Taylor–Marr center, X(357): The first Morley triangle is perspective to triangle ${\displaystyle \triangle ABC}$:[7] the lines each connecting a vertex of the original triangle with the opposite vertex of the Morley triangle concur at the point

$${\displaystyle \sec {\tfrac {1}{3}}A\,:\,\sec {\tfrac {1}{3}}B\,:\,\sec {\tfrac {1}{3}}C}$$

• Angle trisection
• Hofstadter points
• Morley centers

• Connes, Alain (1998), "A new proof of Morley's theorem", Publications Mathématiques de l'IHÉS, S88: 43–46.
• Connes, Alain (December 2004), "Symmetries" (PDF), European Mathematical Society Newsletter, 54.
• Coxeter, H. S. M.; Greitzer, S. L. (1967), Geometry Revisited, The Mathematical Association of America, LCCN 67-20607
• Francis, Richard L. (2002), "Modern Mathematical Milestones: Morley's Mystery" (PDF), Missouri Journal of Mathematical Sciences, 14 (1), doi:10.35834/2002/1401016.
• Guy, Richard K. (2007), "The lighthouse theorem, Morley & Malfatti—a budget of paradoxes" (PDF), American Mathematical Monthly, 114 (2): 97–141, doi:10.1080/00029890.2007.11920398, JSTOR 27642143, MR 2290364, S2CID 46275242, archived from the original (PDF) on 2010-04-01.
• Oakley, C. O.; Baker, J. C. (1978), "The Morley trisector theorem", American Mathematical Monthly, 85 (9): 737–745, doi:10.2307/2321680, JSTOR 2321680, S2CID 56066204.
• Taylor, F. Glanville; Marr, W. L. (1913–14), "The six trisectors of each of the angles of a triangle", Proceedings of the Edinburgh Mathematical Society, 33: 119–131, doi:10.1017/S0013091500035100.

• Morleys Theorem at MathWorld
• Morley's Trisection Theorem at MathPages
• Morley's Theorem by Oleksandr Pavlyk, The Wolfram Demonstrations Project.