{4} a^2
| angle = 60°
}}
In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, an
equilateral triangle is a
triangleA 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 ....
in which all three sides are equal. In traditional or
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, equilateral triangles are also
equiangularIn Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon.The only equiangular triangle is the equilateral triangle...
; that is, all three internal angles are also congruent to each other and are each 60°. They are
regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
s, and can therefore also be referred to as regular triangles.
Characterizations
A triangle that has the sides
a, b, c, and where
R and
r are the radii of the circumcircle and
incircleIn geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides...
respectively, is equilateral
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it satisfies any one of the following necessary and sufficient conditions:
- the circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter....
coincides with the incenterIn geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides...
.
- the circumcenter coincides with the orthocenter
In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...
.
- the circumcenter coincides with the centroid
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...
.
- the circumcenter coincides with the Nagel point
In geometry, the Nagel point is a point associated with any triangle. Given a triangle ABC, let TA, TB, and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively...
.
- the incenter coincides with the nine-point center.
- it is equiangular.
- the three altitude
In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...
s have equal lengths.
- the three median
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...
s have equal lengths.
- the three angle bisectors have equal lengths.
- the three exradii are equal.
Properties
Assuming the lengths of the sides of the equilateral triangle are
, we can determine using the
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
that:
By
Euler's inequalityIn geometry, Euler's theorem, named after Leonhard Euler, states that the distance d between the circumcentre and incentre of a triangle can be expressed as d^2=R \,...
, the equilateral triangle has the smallest ratio
R/
r of the circumradius to the inradius of any triangle: specifically,
R/
r =2.
The ratio of the area to the square of the perimeter of an equilateral triangle,
is larger than that for any other triangle.
For any point P in the plane, with distances
p,
q, and
t from the vertices A, B, and C respectively,
.
For any interior point P in an equilateral triangle, with distances
d,
e, and
f from the sides,
d+e+f = the altitude of the triangle, independent of the location of P.
For any point P on the inscribed circle of an equilateral triangle, with distances
p,
q, and
t from the vertices,
and
.
For any point P on the minor arc BC of the circumcircle, with distances
p,
q, and
t from A, B, and C respectively,
and
moreover, if point D on side BC divides PA into segments PD and DA with DA having length
z and PD having length
y, then
which also equals
if
t ≠
q; and
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. The triangle of greatest area among all those with a given perimeter is equilateral.
An equilateral triangle is the most symmetrical triangle, having 3 lines of
reflectionReflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...
and
rotational symmetryGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
of order 3 about its center.
Its
symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
is the
dihedral group of order 6The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 and the symmetric group of degree 3, with notation S3....
D3.
Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform
polyhedraIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
. Three of the five
Platonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s are composed of equilateral triangles. In particular, the
regular tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
has four equilateral triangles for faces and can be considered the three dimensional analogue of the shape. The plane can be
tiledPlane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in Harmonices Mundi.- Regular tilings :...
using equilateral triangles giving the
triangular tiling.
A result finding an equilateral triangle associated to any triangle is
Morley's trisector theoremIn 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...
.
Geometric construction
An equilateral triangle is easily constructed using a compass.
Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment.
Repeat with the other side of the line.
Finally, connect the point where the two arcs intersect with each end of the line segment
Alternate method:
Draw a circle with radius
r, place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.
The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of
Euclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
.
In culture and society
Equilateral triangles have frequently appeared in man made constructions:
- Some archaeological site
An archaeological site is a place in which evidence of past activity is preserved , and which has been, or may be, investigated using the discipline of archaeology and represents a part of the archaeological record.Beyond this, the definition and geographical extent of a 'site' can vary widely,...
s have equilateral triangles as part of their construction, for example Lepenski VirLepenski Vir is an important Mesolithic archaeological site located in Serbia in the central Balkan peninsula. It consists of one large settlement with around ten satellite villages. The evidence suggests the first human presence in the locality around 7000 BC with the culture reaching its peak...
in Serbia.
- The shape also occurs in modern architecture such as Randhurst Mall
Randhurst Mall, previously known as Randhurst Center or simply Randhurst, was a shopping mall that was located at the corner of Rand Road and Elmhurst Road in Mount Prospect, Illinois. The mall took its name from combining the names of these two roads...
and the Jefferson National Expansion MemorialThe Jefferson National Expansion Memorial is in St. Louis, Missouri, near the starting point of the Lewis and Clark Expedition. It was designated as a National Memorial by Executive Order 7523, on December 21, 1935, and is maintained by the National Park Service .The park was established to...
.
- The Flag of the Philippines
The national flag of the Philippines is a horizontal flag bicolor with equal bands of royal blue and scarlet red, and with a white equilateral triangle at the hoist; in the center of the triangle is a golden yellow sun with eight primary rays, each containing three individual rays, which represent...
, the Seal of the President of the PhilippinesThe Seal of the President of the Philippines is a symbol used to represent the history and dignity of the President of the Philippines. Its original form was designed by Captain Galo B. Ocampo, secretary of the Philippine Heraldry Committee, and patterned after the Seal of the President of the...
and the Flag of JunqueirópolisThe flag of Junqueirópolis is the official flag of the municipality of Junqueirópolis in the western region of the state of São Paulo, Brazil. It was legally instituted on 30 May 1978....
contain equilateral triangles.
- The shape has been given mystical significance, as a representation of the trinity
The Christian doctrine of the Trinity defines God as three divine persons : the Father, the Son , and the Holy Spirit. The three persons are distinct yet coexist in unity, and are co-equal, co-eternal and consubstantial . Put another way, the three persons of the Trinity are of one being...
in The Two BabylonsThe Two Babylons is an anti-Catholic religious pamphlet produced initially by the Scottish theologian and Presbyterian Alexander Hislop in 1853. It was later expanded in 1858 and finally published as a book in 1919...
and forming part of the tetractysThe tetractys , or tetrad, is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row...
figure used by the PythagoreansPythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
.
- It is a shape of a variety of road signs
Traffic signs or road signs are signs erected at the side of roads to provide information to road users. With traffic volumes increasing over the last eight decades, many countries have adopted pictorial signs or otherwise simplified and standardized their signs to facilitate international travel...
, including the Yield signIn road transport, a ' or ' traffic sign indicates that each driver must prepare to stop if necessary to let a driver on another approach proceed. A driver who stops has yielded the right of way to another...
.
- Tau Kappa Epsilon
Tau Kappa Epsilon is a college fraternity founded on January 10, 1899 at Illinois Wesleyan University with chapters in the United States, and Canada, and affiliation with a German fraternity system known as the Corps of the Weinheimer Senioren Convent...
a NIC- Banking and insurance companies:* National Insurance Company Limited, India.* National Insurance Company, Pakistan.* National Insurance Corporation, Kenya.* National Insurance Corporation, Tanzania.* National Insurance Corporation, Uganda....
FraternityA fraternity is a brotherhood, though the term usually connotes a distinct or formal organization. An organization referred to as a fraternity may be a:*Secret society*Chivalric order*Benefit society*Friendly society*Social club*Trade union...
uses the Equilateral triangle as its primary symbol.
See also
- Dragon's Eye (symbol)
The Dragon's Eye is an ancient Germanic symbol as discovered by Rudolf Koch. The Dragon's Eye is an equilateral triangle pointing downward with a "Y" in the middle connecting the three points of the triangle together. According to Carl G...
- 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...
- Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from a point to the sides of an equilateral triangle equals the length of the triangle's altitude....
- Almost-equilateral Heronian triangle