In geometry, an equilateral triangle is a triangle in which all three sides have equal lengths. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position 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 In traditional or Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also equal to each other and are each 60°. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Euclidean geometry, an equiangular polygon is a Polygon whose vertex angles are equal They are regular polygons, and can therefore also be referred to as regular triangles. General properties These properties apply to both convex and star regular polygons

Properties

The area of an equilateral triangle with sides of length is and its altitude is . In Geometry, an altitude of a triangle is a Straight line through a vertex and Perpendicular to (i These formulas can be derived using the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Its symmetry group is the dihedral group of order 6 D₃. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is The smallest Non-abelian group has 6 elements It is a Dihedral group with notation D 3 or D 6

Regular tetrahedron is made of four equilateral triangles.

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 polyhedra. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions Three of the five Platonic solids are composed of equilateral triangles. In Geometry, a Platonic solid is a convex Regular polyhedron. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three dimensional analogue of the shape. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. The plane can be tiled using equilateral triangles giving the triangular tiling. Plane tilings by Regular polygons have been widely used since antiquity In Geometry, the triangular tiling is one of the three regular tilings of the Euclidean plane.

A result finding an equilateral triangle associated to any triangle is Morley's trisector theorem. 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

Geometric construction

Construction of equilateral triangle with compass and straightedge

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 past halfway of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intesect 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 centres of the circles and either of the points of intersection.

Almost-equilateral Heronian triangles

A Heronian triangle is a triangle with rational sides and rational area. In Geometry, a Heronian triangle is a Triangle whose sidelengths and area are all Rational numbers It is named after Hero of Alexandria Since the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides, expressed as integers, are of the form n − 1, n, n + 1. The first few examples of these almost-equilateral triangles are set forth in the following table.

Subsequent values of n can be found by multiplying the last known value by 4, then subtracting the next to the last one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc), as expressed in

This sequence can also be generated from the solutions to the Pell equation x² − 3y² = 1, which can in turn be derived from the regular continued fraction expansion for √3. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} ^[1]

In culture and society

Equilateral triangles have frequently appeared in man made constructions:

• Some archaeological sites have equilateral triangles as part of their construction, for example Lepenski Vir in Serbia. An archaeological site is a place (or group of physical sites in which evidence of past activity is preserved (either Prehistoric or historic or contemporary and Lepenski Vir (Лепенски Вир is an important Mesolithic Archaeological site located in Serbia in the central Balkan Peninsula
• The shape also occurs in modern architecture such as Randhurst Mall and the Jefferson National Expansion Memorial. Randhurst Mall was a Shopping mall that originally opened in 1962 The Jefferson National Expansion Memorial is located in St Louis Missouri near
• The Seal of the President of the Philippines and Flag of Junqueirópolis contain equilateral triangles. The Seal of the President of the Philippines is a symbol used to represent the history and dignity of the president of the Philippines. The flag of Junqueirópolis ( bandeira de Junqueirópolis) is the official Flag of the Municipality of Junqueirópolis in the western region of
• The shape has been given mystical significance, as a representation of the trinity in The Two Babylons and forming part of the tetractys figure used by the Pythagoreans. SSC RF "Troitsk Institute of Innovative and Termonuclear Research" or TRINITY for shprt Троицкий Институт инновационных и термоядерных The Two Babylons was an Anti-Catholic religious Pamphlet produced initially by the Scottish theologian and Protestant Presbyterian The Tetractys is a triangular figure consisting of ten points arranged in four rows one two three and four points in each row Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced

See also

• Trigonometry
• Viviani's theorem

References

1. ^ Takeaki Murasaki (2004), On the Heronian Triple (n+1, n, n−1), Sci. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Viviani's theorem is the theorem named after Vincenzo Viviani stating that the sum of the distances from a point to the sides of an Equilateral triangle equals the Rep. Fac. Educ. , Gunma Univ. 52, 9-15.

External links

• MathWorld - an overview of the Euclidean construction of an equilateral triangle