
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
::
The last equation can be verified by applying the sum of two angles identity to the left side twice and eliminating the cosine.
Points
are constructed on
as shown. We have
, the sum of any triangle's angles, so
Therefore, the angles of triangle
are
and
From the figure
and
Also from the figure
::
and
The law of sines applied to triangles
and
yields
and
Express the height of triangle
in two ways
::
and
::
where equation (1) was used to replace
and
in these two equations. Substituting equations (2) and (5) in the
equation and equations (3) and (6) in the
equation gives
::
and
::
Since the numerators are equal
::
or
::
Since angle
and angle
are equal and the sides forming these angles are in the same ratio, triangles
and
are similar.
Similar angles
and
equal
, and similar angles
and
equal
Similar arguments yield the base angles of triangles
and
In particular angle
is found to be
and from the figure we see that
::
Substituting yields
::
where equation (4) was used for angle
and therefore
::
Similarly the other angles of triangle
are found to be
Side and area
The first Morley triangle has side lengths
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
the area of Morley's triangle can be expressed as
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:
Another of Morley's equilateral triangles that is also a central triangle is called the second Morley triangle and is given by these vertices:
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:
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:
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:
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
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
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 Theoremat MathWorld
at MathPages
Morley's Theoremby 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