Morley's trisector theorem
Encyclopedia
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 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. Morley's Theorem is one of the most surprising and beautiful theorems in plane geometry
.
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 algebra
ic proof by 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. Interestingly, Morley's theorem does not hold in spherical
and hyperbolic geometry
.
Arguably the best 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.
The result can also be proved by the technique of reverse reconstruction
.
relative to a triangle ABC as follows:
Another of Morley's equilateral triangle that is also 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 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 others, is obtained by rotating the circumtangential triangle π/6 about its center. Called the circumnormal triangle, its vertices are as follows:
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
.
of the first Morley triangle is given by
The first Morley triangle is perspective to triangle ABC, and the perspector is the point
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
, Morley's trisector theorem states that in any triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
, the three points of intersection of the adjacent angle trisectors
Angle trisection
Angle trisection is a classic problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one-third of a given arbitrary angle, using only two tools: an un-marked straightedge, and a compass....
form an equilateral triangle, called the Morley triangle. The theorem was discovered in 1899 by Anglo-American
English American
English Americans are citizens or residents of the United States whose ancestry originates wholly or partly in England....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Frank Morley
Frank Morley
Frank Morley was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry...
. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles. Morley's Theorem is one of the most surprising and beautiful theorems in plane geometry
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
.
Proofs
There are many proofsMathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
of Morley's theorem, some of which are very technical.
Several early proofs were based on delicate trigonometric
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
calculations. The first published geometric proof was given by M.T.Naraniengar in 1909. Recent proofs include an algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
ic proof by extending the theorem to general fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and John Conway
John Conway
John Conway may refer to:* John Horton Conway, mathematician at Princeton University. Popularly known for Conway's Game of Life* John B. Conway, mathematician, functional analyst, George Washington University...
'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
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
to any selected triangle. Interestingly, Morley's theorem does not hold in spherical
Spherical geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
and hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
Arguably the best proof uses the trigonometric identity :
- sin 3θ ≡ 4 sin θ sin(60°+θ) sin(120°+θ).
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
- h = AB sin 3β = 4AB.AC.DX sin β sin γ / (XE.AY)
- = AC sin 3γ = 4AC.AB.DX sin γ sin β / (XF.AZ).
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.
The result can also be proved by the technique of reverse reconstruction
Reverse reconstruction
Reverse reconstruction is a technique used to solve geometry problems. The basic structure of such a proof goes as follows: Let the problem be to prove that P implies Q. Then show that Q implies P and that the construction of the original diagram is unique. Thus P implies Q. This method of proof...
.
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 coordinatesTrilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...
relative to a triangle ABC as follows:
- A-vertex = 1 : 2 cos(C/3) : 2 cos(B/3)
- B-vertex = 2 cos(C/3) : 1 : 2 cos(A/3)
- C-vertex = 2 cos(B/3) : 2 cos(A/3) : 1
Another of Morley's equilateral triangle that is also central triangle is called the second Morley triangle and is given by these vertices:
- A-vertex = 1 : 2 cos(C/3 − 2π/3) : 2 cos(B/3 − 2π/3)
- B-vertex = 2 cos(C/3 − 2π/3) : 1 : 2 cos(A/3 − 2π/3)
- C-vertex = 2 cos(B/3 − 2π/3) : 2 cos(A/3 − 2π/3) : 1
The third of Morley's 18 equilateral triangles that is also central triangle is called the third Morley triangle and is given by these vertices:
- A-vertex = 1 : 2 cos(C/3 − 4π/3) : 2 cos(B/3 − 4π/3)
- B-vertex = 2 cos(C/3 − 4π/3) : 1 : 2 cos(A/3 − 4π/3)
- C-vertex = 2 cos(B/3 − 4π/3) : 2 cos(A/3 − 4π/3) : 1
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
Isogonal conjugate
In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P...
of X. This equilateral triangle, called the circumtangential triangle, has these vertices:
- A-vertex = csc(C/3 − B/3) : csc(B/3 + 2C/3) : −csc(C/3 + 2B/3)
- B-vertex = −csc(A/3 + 2C/3) : csc(A/3 − C/3) : csc(C/3 + 2A/3)
- C-vertex = csc(A/3 + 2B/3) : −csc(B/3 + 2A/3) : csc(B/3 − A/3)
A fifth equilateral triangle, also homothetic to others, is obtained by rotating the circumtangential triangle π/6 about its center. Called the circumnormal triangle, its vertices are as follows:
- A-vertex = sec(C/3 − B/3) : −sec(B/3 + 2C/3) : −sec(C/3 + 2B/3)
- B-vertex = −sec(A/3 + 2C/3) : sec(A/3 − C/3) : −sec(C/3 + 2A/3)
- C-vertex = −sec(A/3 + 2B/3) : −sec(B/3 + 2A/3) : sec(B/3 − A/3)
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
Pappus graph
In the mathematical field of graph theory, the Pappus graph is a 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon...
.
Related triangle centers
The centroidCentroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
of the first Morley triangle is given by
- Morley center = X(356) = cos(A/3) + 2 cos(B/3)cos(C/3) : cos(B/3) + 2 cos(C/3)cos(A/3) : cos(C/3) + 2 cos(A/3)cos(B/3)
The first Morley triangle is perspective to triangle ABC, and the perspector is the point
- 1st Morley–Taylor–Marr center = X(357) = sec(A/3) : sec(B/3) : sec(C/3)
External links
- Morleys Theorem at MathWorld
- Morley's Trisection Theorem at MathPages
- Morley's Theorem by Oleksandr Pavlyk, The Wolfram Demonstrations Project.