Editing Morley's trisector theorem

{{short description|3 intersections of any triangle's adjacent angle trisectors form an equilateral triangle}}
[[File:Morley triangle.svg|right|frame|If each vertex angle of the outer triangle is trisected, Morley's trisector theorem states that the purple triangle will be equilateral.]]
In [[plane geometry]], '''Morley's trisector theorem''' states that in any [[triangle]], the three points of intersection of the adjacent [[Angle trisection|angle trisectors]] form an [[equilateral triangle]], called the '''first Morley triangle''' or simply the '''Morley triangle'''. The theorem was discovered in 1899 by [[English American|Anglo-American]] [[mathematician]] [[Frank Morley]]. It has various generalizations; in particular, if all the trisectors are intersected, one obtains four other equilateral triangles.

==Proofs==
There are many [[Mathematical proof|proofs]] of Morley's theorem, some of which are very technical.<ref>{{citation|url=http://www.cut-the-knot.org/triangle/Morley/index.shtml|title=Morley's Miracle|publisher=[[Cut-the-knot]]|last=Bogomolny|first=Alexander|authorlink= Alexander Bogomolny |accessdate=2010-01-02}}</ref>  
Several early proofs were  based on delicate [[trigonometry|trigonometric]] calculations. Recent proofs include an [[algebra]]ic proof by {{harvs|first=Alain|last=Connes|authorlink=Alain Connes|txt|year=1998|year2=2004}} extending the theorem to general [[Field (mathematics)|fields]] other than characteristic three, and [[John Horton Conway|John Conway]]'s elementary geometry proof.<ref>{{Citation |last=Bogomolny |first=Alexander |title=J. Conway's proof |url=http://www.cut-the-knot.org/triangle/Morley/conway.shtml |publisher=[[Cut-the-knot]] |access-date=2021-12-03 |authorlink=Alexander Bogomolny}}</ref><ref>{{citation|chapter-url=http://thewe.net/math/conway.pdf|title=Power|editor1-last=Blackwell|editor1-first=Alan|editor2-last=Mackay|editor2-first=David|editor2-link=David J. C. MacKay|year=2006|chapter=The Power of Mathematics|last=Conway|first=John|author-link=John Horton Conway|publisher=Cambridge University Press|accessdate=2010-10-08|pages=36–50|isbn=978-0-521-82377-7}}</ref>  The latter starts with an equilateral triangle and shows that a triangle may be built around it which will be [[Similarity (geometry)|similar]] to any selected triangle.  Morley's theorem does not hold in [[spherical geometry|spherical]]<ref>[http://lienhard-wimmer.com/applets/dreieck/Morley.html Morley's Theorem in Spherical Geometry], [[Java applet]].</ref> and [[hyperbolic geometry]].

[[File:Morley Proof.svg|thumb|right|480px|Fig 1. &nbsp; Elementary proof of Morley's trisector theorem]]
One proof uses the trigonometric identity

{{NumBlk|::|<math>\sin(3\theta)=4\sin\theta\sin(60^\circ+\theta)\sin(120^\circ+\theta)</math>|{{EquationRef|1}}}}

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

::<math>\sin(3\theta)=-4\sin^3\theta+3\sin\theta.</math>

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

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

From the figure

{{NumBlk|::|<math>\sin(60^\circ+\beta)=\frac{\overline{DX}}{\overline{XE}}</math>|{{EquationRef|2}}}}

and

{{NumBlk|::|<math>\sin(60^\circ+\gamma)=\frac{\overline{DX}}{\overline{XF}}.</math>|{{EquationRef|3}}}}

Also from the figure

::<math>\angle{AYC}=180^\circ-\alpha-\gamma=120^\circ+\beta</math>

and

{{NumBlk|::|<math>\angle{AZB}=120^\circ+\gamma.</math>|{{EquationRef|4}}}}

The law of sines applied to triangles <math>AYC</math> and <math>AZB</math> yields

{{NumBlk|::|<math>\sin(120^\circ+\beta)=\frac{\overline{AC}}{\overline{AY}}\sin\gamma</math>|{{EquationRef|5}}}}

and

{{NumBlk|::|<math>\sin(120^\circ+\gamma)=\frac{\overline{AB}}{\overline{AZ}}\sin\beta.</math>|{{EquationRef|6}}}}

Express the height of triangle <math>ABC</math> in two ways

::<math>h=\overline{AB} \sin(3\beta)=\overline{AB}\cdot 4\sin\beta\sin(60^\circ+\beta)\sin(120^\circ+\beta)</math>

and

::<math>h=\overline{AC} \sin(3\gamma)=\overline{AC}\cdot 4\sin\gamma\sin(60^\circ+\gamma)\sin(120^\circ+\gamma).</math>

where equation (1) was used to replace <math>\sin(3\beta)</math> and <math>\sin(3\gamma)</math> in these two equations. Substituting equations (2) and (5) in the <math>\beta</math> equation and equations (3) and (6) in the <math>\gamma</math> equation gives

::<math>h=4\overline{AB}\sin\beta\cdot\frac{\overline{DX}}{\overline{XE}}\cdot\frac{\overline{AC}}{\overline{AY}}\sin\gamma</math>

and

::<math>h=4\overline{AC}\sin\gamma\cdot\frac{\overline{DX}}{\overline{XF}}\cdot\frac{\overline{AB}}{\overline{AZ}}\sin\beta</math>

Since the numerators are equal

::<math>\overline{XE}\cdot\overline{AY}=\overline{XF}\cdot\overline{AZ}</math>

or

::<math>\frac{\overline{XE}}{\overline{XF}}=\frac{\overline{AZ}}{\overline{AY}}.</math>

Since angle <math>EXF</math> and angle <math>ZAY</math> are equal and the sides forming these angles are in the same ratio, triangles <math>XEF</math> and <math>AZY</math> are similar. 

Similar angles <math>AYZ</math> and <math>XFE</math> equal <math>(60^\circ+\gamma)</math>, and similar angles <math>AZY</math> and <math>XEF</math> equal <math>(60^\circ+\beta).</math> Similar arguments yield the base angles of triangles <math>BXZ</math> and <math>CYX.</math>

In particular angle <math>BZX</math> is found to be <math>(60^\circ+\alpha)</math> and from the figure we see that

::<math>\angle{AZY}+\angle{AZB}+\angle{BZX}+\angle{XZY}=360^\circ.</math>

Substituting yields

::<math>(60^\circ+\beta)+(120^\circ+\gamma)+(60^\circ+\alpha)+\angle{XZY}=360^\circ</math>

where equation (4) was used for angle <math>AZB</math> and therefore

::<math>\angle{XZY}=60^\circ.</math>

Similarly the other angles of triangle <math>XYZ</math> are found to be <math>60^\circ.</math>

==Side and area==

The first Morley triangle has side lengths<ref>{{MathWorld |id= FirstMorleyTriangle|title=First Morley Triangle |access-date=2021-12-03}}</ref>

<math display=block>
a^\prime=b^\prime=c^\prime=8R\,\sin\tfrac13A\,\sin\tfrac13B\,\sin\tfrac13C,
</math>

where ''R'' is the [[circumradius]] of the original triangle and ''A, B,'' and ''C'' are the angles of the original triangle. Since the [[area (geometry)|area]] of an equilateral triangle is <math>\tfrac{\sqrt{3}}{4}a'^2,</math> the area of Morley's triangle can be expressed as

<math display=block>
\text{Area} = 16 \sqrt{3}R^2\, \sin^2\!\tfrac13A\, \sin^2\!\tfrac13B\, \sin^2\!\tfrac13C.
</math>

==Morley's triangles==
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:

<math display=block>\begin{array}{ccccccc}
  A \text{-vertex} &=& 1 &:& 2 \cos\tfrac13 C &:& 2 \cos\tfrac13 B \\[5mu]
  B \text{-vertex} &=& 2 \cos\tfrac13 C &:& 1 &:& 2 \cos\tfrac13 A \\[5mu]
  C \text{-vertex} &=& 2 \cos\tfrac13 B &:& 2 \cos\tfrac13 A &:& 1
\end{array}</math>

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

<math display=block>\begin{array}{ccccccc}
  A \text{-vertex} &=& 1 &:& 2 \cos\tfrac13(C - 2\pi) &:& 2 \cos\tfrac13(B - 2\pi) \\[5mu]
  B \text{-vertex} &=& 2 \cos\tfrac13(C - 2\pi) &:& 1 &:& 2 \cos\tfrac13(A - 2\pi) \\[5mu]
  C \text{-vertex} &=& 2 \cos\tfrac13(B - 2\pi) &:& 2 \cos\tfrac13(A - 2\pi) &:& 1
\end{array}</math>

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:

<math display=block>\begin{array}{ccccccc}
  A \text{-vertex} &=& 1 &:& 2 \cos\tfrac13(C + 2\pi) &:& 2 \cos\tfrac13(B + 2\pi) \\[5mu]
  B \text{-vertex} &=& 2 \cos\tfrac13(C + 2\pi) &:& 1 &:& 2 \cos\tfrac13(A + 2\pi) \\[5mu]
  C \text{-vertex} &=& 2 \cos\tfrac13(B + 2\pi) &:& 2 \cos\tfrac13(A + 2\pi) &:& 1
\end{array}</math>

The first, second, and third Morley triangles are pairwise [[Homothetic transformation|homothetic]].  Another homothetic triangle is formed by the three points ''X'' on the circumcircle of triangle ''ABC'' at which the line ''XX''<sup>&nbsp;&minus;1</sup> is tangent to the circumcircle, where ''X''<sup>&nbsp;&minus;1</sup> denotes the [[isogonal conjugate]] of ''X''.  This equilateral triangle, called the '''circumtangential triangle''', has these vertices:

<math display=block>\begin{array}{lllllll}
  A \text{-vertex} &=& \phantom{-}\csc\tfrac13(C - B) &:& \phantom{-}\csc\tfrac13(2C + B) &:& -\csc\tfrac13(C + 2B) \\[5mu]
  B \text{-vertex} &=& -\csc\tfrac13(A + 2C) &:& \phantom{-}\csc\tfrac13(A - C) &:& \phantom{-}\csc\tfrac13(2A + C) \\[5mu]
  C \text{-vertex} &=& \phantom{-}\csc\tfrac13(2B + A) &:& -\csc\tfrac13(B + 2A) &:& \phantom{-}\csc\tfrac13(B - A)
\end{array}</math>

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

<math display=block>\begin{array}{lllllll}
  A \text{-vertex} &=& \phantom{-}\sec\tfrac13(C - B) &:& -\sec\tfrac13(2C + B) &:& -\sec\tfrac13(C + 2B) \\[5mu]
  B \text{-vertex} &=& -\sec\tfrac13(A + 2C) &:& \phantom{-}\sec\tfrac13(A - C) &:& -\sec\tfrac13(2A + C) \\[5mu]
  C \text{-vertex} &=& -\sec\tfrac13(2B + A) &:& -\sec\tfrac13(B + 2A) &:& \phantom{-}\sec\tfrac13(B - A)
\end{array}</math>

An operation called "[[triangle extraversion|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]].<ref>{{harvtxt|Guy|2007}}.</ref>

==Related triangle centers==

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

<math display=block>
\cos\tfrac13A + 2\cos\tfrac13B\,\cos\tfrac13C \,:\, \cos\tfrac13B + 2\cos\tfrac13C\,\cos\tfrac13A \,:\, \cos\tfrac13C + 2\cos\tfrac13A\,\cos\tfrac13B
</math>

'''1st Morley–Taylor–Marr center''', ''X''(357): The first Morley triangle is [[perspective (geometry)|perspective]] to triangle {{nobr|<math>\triangle ABC</math>{{sfn|Taylor Marr|1913}} :<ref>Fox, M. D.; and Goggins, J. R. "Morley's diagram generalised", ''[[Mathematical Gazette]]'' 87, November 2003, 453–467.</ref>}}<< the lines each connecting a vertex of the original triangle with the opposite vertex of the Morley triangle [[concurrent lines|concur]] at the point

<math display=block>
\sec\tfrac13A \,:\, \sec\tfrac13B \,:\, \sec\tfrac13C
</math>

==See also==
*[[Angle trisection]]
*[[Hofstadter points]]
*[[Morley centers]]

== Notes ==
{{reflist|2}}

== References ==
*{{citation|first=Alain|last=Connes|authorlink=Alain Connes|url=http://www.numdam.org/item?id=PMIHES_1998__S88__43_0|title=A new proof of Morley's theorem|journal=Publications Mathématiques de l'IHÉS|volume=S88|year=1998|pages=43–46}}.
*{{citation|first=Alain|last=Connes|authorlink=Alain Connes|url=http://www.ems-ph.org/journals/newsletter/pdf/2004-12-54.pdf|title=Symmetries|journal=European Mathematical Society Newsletter|volume=54|date=December 2004}}.
*{{citation|first1=H. S. M.|last1=Coxeter|authorlink=Coxeter|first2=S. L.|last2=Greitzer|title=Geometry Revisited|publisher=[[The Mathematical Association of America]]|year=1967|lccn=67-20607}}
*{{citation|first=Richard L.|last=Francis|url=http://cs.ucmo.edu/~mjms/2002.1/francis9.pdf|title=Modern Mathematical Milestones: Morley's Mystery|journal=Missouri Journal of Mathematical Sciences|volume=14|issue=1|year=2002|doi=10.35834/2002/1401016|doi-access=free}}.
*{{citation
 |last        = Guy
 |first       = Richard K.
 |authorlink  = Richard K. Guy
 |mr          = 2290364
 |issue       = 2
 |journal     = [[American Mathematical Monthly]]
 |pages       = 97–141
 |title       = The lighthouse theorem, Morley & Malfatti—a budget of paradoxes
 |jstor       = 27642143
 |url         = http://www.math.ucalgary.ca/files/publications/3414848.pdf
 |volume      = 114
 |year        = 2007
 |url-status     = dead
 |archiveurl  = https://web.archive.org/web/20100401030732/http://www.math.ucalgary.ca/files/publications/3414848.pdf
 |archivedate = 2010-04-01 |doi=10.1080/00029890.2007.11920398
|s2cid = 46275242
 }}.
*{{citation|doi=10.2307/2321680|first1=C. O.|last1=Oakley|first2=J. C.|last2=Baker|title=The Morley trisector theorem|journal=[[American Mathematical Monthly]]|volume=85|issue=9 |year=1978|pages=737–745|jstor=2321680|s2cid=56066204 }}.

*{{citation|first1=F. Glanville|last1=Taylor|first2=W. L.|last2=Marr|title=The six trisectors of each of the angles of a triangle|journal=Proceedings of the Edinburgh Mathematical Society|volume=33|year=1913–14|pages=119–131|doi=10.1017/S0013091500035100|doi-access=free|ref={{harvid|Taylor Marr|1913}}}}.

== External links ==
* [http://mathworld.wolfram.com/MorleysTheorem.html Morleys Theorem] at MathWorld
* [http://www.mathpages.com/home/kmath376/kmath376.htm Morley's Trisection Theorem] at MathPages
* [http://demonstrations.wolfram.com/MorleysTheorem/ Morley's Theorem] by Oleksandr Pavlyk, [[The Wolfram Demonstrations Project]].

[[Category:Theorems about triangles]]