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In
plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, Morley's trisector theorem states that in any
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, the three points of intersection of the adjacent angle trisectors form an
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Frank Morley Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celeb ...
. It has various generalizations; in particular, if all the trisectors are intersected, one obtains four other equilateral triangles.


Proofs

There are many proofs of Morley's theorem, some of which are very technical. Several early proofs were based on delicate trigonometric calculations. Recent proofs include an
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
ic proof by extending the theorem to general
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
other than characteristic three, 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
. One proof uses the trigonometric identity which, by using of the sum of two angles identity, can be shown to be equal to ::\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 D, E, F are constructed on \overline as shown. We have 3\alpha+3\beta+3\gamma=180^\circ, the sum of any triangle's angles, so \alpha+\beta+\gamma=60^\circ. Therefore, the angles of triangle XEF are \alpha, (60^\circ+\beta), and (60^\circ+\gamma). From the figure and Also from the figure ::\angle=180^\circ-\alpha-\gamma=120^\circ+\beta and The law of sines applied to triangles AYC and AZB yields and Express the height of triangle ABC in two ways ::h=\overline \sin(3\beta)=\overline\cdot 4\sin\beta\sin(60^\circ+\beta)\sin(120^\circ+\beta) and ::h=\overline \sin(3\gamma)=\overline\cdot 4\sin\gamma\sin(60^\circ+\gamma)\sin(120^\circ+\gamma). where equation (1) was used to replace \sin(3\beta) and \sin(3\gamma) in these two equations. Substituting equations (2) and (5) in the \beta equation and equations (3) and (6) in the \gamma equation gives ::h=4\overline\sin\beta\cdot\frac\cdot\frac\sin\gamma and ::h=4\overline\sin\gamma\cdot\frac\cdot\frac\sin\beta Since the numerators are equal ::\overline\cdot\overline=\overline\cdot\overline or ::\frac=\frac. Since angle EXF and angle ZAY are equal and the sides forming these angles are in the same ratio, triangles XEF and AZY are similar. Similar angles AYZ and XFE equal (60^\circ+\gamma), and similar angles AZY and XEF equal (60^\circ+\beta). Similar arguments yield the base angles of triangles BXZ and CYX. In particular angle BZX is found to be (60^\circ+\alpha) and from the figure we see that ::\angle+\angle+\angle+\angle=360^\circ. Substituting yields ::(60^\circ+\beta)+(120^\circ+\gamma)+(60^\circ+\alpha)+\angle=360^\circ where equation (4) was used for angle AZB and therefore ::\angle=60^\circ. Similarly the other angles of triangle XYZ are found to be 60^\circ.


Side and area

The first Morley triangle has side lengths a^\prime=b^\prime=c^\prime=8R\,\sin\tfrac13A\,\sin\tfrac13B\,\sin\tfrac13C, where ''R'' is the circumradius of the original triangle and ''A, B,'' and ''C'' are the angles of the original triangle. Since the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of an equilateral triangle is \tfraca'^2, the area of Morley's triangle can be expressed as \text = 16 \sqrtR^2\, \sin^2\!\tfrac13A\, \sin^2\!\tfrac13B\, \sin^2\!\tfrac13C.


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: \begin A \text &=& 1 &:& 2 \cos\tfrac13 C &:& 2 \cos\tfrac13 B \\ mu B \text &=& 2 \cos\tfrac13 C &:& 1 &:& 2 \cos\tfrac13 A \\ mu C \text &=& 2 \cos\tfrac13 B &:& 2 \cos\tfrac13 A &:& 1 \end Another of Morley's equilateral triangles that is also a central triangle is called the second Morley triangle and is given by these vertices: \begin A \text &=& 1 &:& 2 \cos\tfrac13(C - 2\pi) &:& 2 \cos\tfrac13(B - 2\pi) \\ mu B \text &=& 2 \cos\tfrac13(C - 2\pi) &:& 1 &:& 2 \cos\tfrac13(A - 2\pi) \\ mu C \text &=& 2 \cos\tfrac13(B - 2\pi) &:& 2 \cos\tfrac13(A - 2\pi) &:& 1 \end 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: \begin A \text &=& 1 &:& 2 \cos\tfrac13(C + 2\pi) &:& 2 \cos\tfrac13(B + 2\pi) \\ mu B \text &=& 2 \cos\tfrac13(C + 2\pi) &:& 1 &:& 2 \cos\tfrac13(A + 2\pi) \\ mu C \text &=& 2 \cos\tfrac13(B + 2\pi) &:& 2 \cos\tfrac13(A + 2\pi) &:& 1 \end 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: \begin A \text &=& \phantom\csc\tfrac13(C - B) &:& \phantom\csc\tfrac13(2C + B) &:& -\csc\tfrac13(C + 2B) \\ mu B \text &=& -\csc\tfrac13(A + 2C) &:& \phantom\csc\tfrac13(A - C) &:& \phantom\csc\tfrac13(2A + C) \\ mu C \text &=& \phantom\csc\tfrac13(2B + A) &:& -\csc\tfrac13(B + 2A) &:& \phantom\csc\tfrac13(B - A) \end A fifth equilateral triangle, also homothetic to the others, is obtained by rotating the circumtangential triangle /6 about its center. Called the circumnormal triangle, its vertices are as follows: \begin A \text &=& \phantom\sec\tfrac13(C - B) &:& -\sec\tfrac13(2C + B) &:& -\sec\tfrac13(C + 2B) \\ mu B \text &=& -\sec\tfrac13(A + 2C) &:& \phantom\sec\tfrac13(A - C) &:& -\sec\tfrac13(2A + C) \\ mu C \text &=& -\sec\tfrac13(2B + A) &:& -\sec\tfrac13(B + 2A) &:& \phantom\sec\tfrac13(B - A) \end An operation called "
extraversion Extraversion and introversion are a central trait dimension in human personality theory. The terms were introduced into psychology by Carl Jung, though both the popular understanding and current psychological usage are not the same as Jung's ...
" 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..


Related triangle centers

The Morley center, ''X''(356),
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
of the first Morley triangle, is given in trilinear coordinates by \cos\tfrac13A + 2\cos\tfrac13B\,\cos\tfrac13C \,:\, \cos\tfrac13B + 2\cos\tfrac13C\,\cos\tfrac13A \,:\, \cos\tfrac13C + 2\cos\tfrac13A\,\cos\tfrac13B 1st Morley–Taylor–Marr center, ''X''(357): The first Morley triangle is perspective to triangle << the lines each connecting a vertex of the original triangle with the opposite vertex of the Morley triangle concur at the point \sec\tfrac13A \,:\, \sec\tfrac13B \,:\, \sec\tfrac13C


See also

*
Angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
* Hofstadter points * Morley centers


Notes


References

*. *. * *. *. *. *{{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


Morleys Theorem
at MathWorld

at MathPages
Morley's Theorem
by Oleksandr Pavlyk,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...
. Theorems about triangles