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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an isosceles triangle () is a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
as a
special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case i ...
. Examples of isosceles triangles include the
isosceles right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
, the
golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...
, and the faces of bipyramids and certain
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan s ...
s. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babyl ...
. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the
pediment Pediments are gables, usually of a triangular shape. Pediments are placed above the horizontal structure of the lintel, or entablature, if supported by columns. Pediments can contain an overdoor and are usually topped by hood moulds. A pedimen ...
s and
gable A gable is the generally triangular portion of a wall between the edges of intersecting roof pitches. The shape of the gable and how it is detailed depends on the structural system used, which reflects climate, material availability, and aesth ...
s of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the
perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of its base. The two angles opposite the legs are equal and are always
acute Acute may refer to: Science and technology * Acute angle ** Acute triangle ** Acute, a leaf shape in the glossary of leaf morphology * Acute (medicine), a disease that it is of short duration and of recent onset. ** Acute toxicity, the adverse eff ...
, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.


Terminology, classification, and examples

Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
defined an isosceles triangle as a triangle with exactly two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides. The difference between these two definitions is that the modern version makes equilateral triangles (with three equal sides) a special case of isosceles triangles. A triangle that is not isosceles (having three unequal sides) is called scalene. "Isosceles" is made from the Greek roots "isos" (equal) and "skelos" (leg). The same word is used, for instance, for
isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined ...
s, trapezoids with two equal sides, and for
isosceles set In discrete geometry, an isosceles set is a set of points with the property that every three of them form an isosceles triangle. More precisely, each three points should determine at most two distances; this also allows degenerate isosceles triang ...
s, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called
legs A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element ...
and the third side is called the base. The angle included by the legs is called the ''vertex angle'' and the angles that have the base as one of their sides are called the ''base angles''. The vertex opposite the base is called the
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, ...
. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle is acute, right or obtuse depends only on the angle at its apex. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, the base angles can not be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right, respectively, an isosceles triangle is obtuse, right or acute if and only if its apex angle is respectively obtuse, right or acute. In Edwin Abbott's book ''
Flatland ''Flatland: A Romance of Many Dimensions'' is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dim ...
'', this classification of shapes was used as a satire of
social hierarchy Social stratification refers to a society's categorization of its people into groups based on socioeconomic factors like wealth, income, race, education, ethnicity, gender, occupation, social status, or derived power (social and political). As ...
: isosceles triangles represented the
working class The working class (or labouring class) comprises those engaged in manual-labour occupations or industrial work, who are remunerated via waged or salaried contracts. Working-class occupations (see also " Designation of workers by collar colou ...
, with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles. As well as the
isosceles right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
, several other specific shapes of isosceles triangles have been studied. These include the Calabi triangle (a triangle with three congruent inscribed squares), the
golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...
and
golden gnomon A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio \varphi to the base side: : = \varphi = \approx 1.618~034~. Angles * The vertex angle is: ::\theta = 2\arcsin = 2\ar ...
(two isosceles triangles whose sides and base are in the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
), the 80-80-20 triangle appearing in the
Langley's Adventitious Angles Langley’s Adventitious Angles is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by Edward Mann Langley in ''The Mathematical Gazette'' in 1922.. The problem In its original form the probl ...
puzzle, and the 30-30-120 triangle of the
triakis triangular tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a he ...
. Five
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan s ...
s, the
triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be se ...
,
triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangula ...
, tetrakis hexahedron,
pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that i ...
, and
triakis icosahedron In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 p ...
, each have isosceles-triangle faces, as do infinitely many
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
s and bipyramids.


Formulas


Height

For any isosceles triangle, the following six
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s coincide: *the
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
, a line segment from the apex perpendicular to the base, *the
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
from the apex to the base, *the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
from the apex to the midpoint of the base, *the
perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of the base within the triangle, *the segment within the triangle of the unique
axis of symmetry Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.
of the triangle, and *the segment within the triangle of the
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...
of the triangle, except when the triangle is
equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
. Their common length is the height h of the triangle. If the triangle has equal sides of length a and base of length b, the general triangle formulas for the lengths of these segments all simplify to :h=\sqrt. This formula can also be derived from the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles. The Euler line of any triangle goes through the triangle's
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
(the intersection of its three altitudes), its
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 surface of the figure. The same definition extends to any ...
(the intersection of its three medians), and its
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
(the intersection of the perpendicular bisectors of its three sides, which is also the center of the circumcircle that passes through the three vertices). In an isosceles triangle with exactly two equal sides, these three points are distinct, and (by symmetry) all lie on the symmetry axis of the triangle, from which it follows that the Euler line coincides with the axis of symmetry. The
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
of the triangle also lies on the Euler line, something that is not true for other triangles. If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles.


Area

The area T of an isosceles triangle can be derived from the formula for its height, and from the general formula for the area of a triangle as half the product of base and height: :T=\frac\sqrt. The same area formula can also be derived from
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
for the area of a triangle from its three sides. However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between the
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate na ...
and side length in those triangles. If the apex angle (\theta) and leg lengths (a) of an isosceles triangle are known, then the area of that triangle is: :T=\fraca^2\sin\theta. This is a special case of the general formula for the area of a triangle as half the product of two sides times the sine of the included angle.


Perimeter

The perimeter p of an isosceles triangle with equal sides a and base b is just :p = 2a + b. As in any triangle, the area T and perimeter p are related by the
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
:p^2>12\sqrtT. This is a strict inequality for isosceles triangles with sides unequal to the base, and becomes an equality for the equilateral triangle. The area, perimeter, and base can also be related to each other by the equation :2pb^3 -p^2b^2 + 16T^2 = 0. If the base and perimeter are fixed, then this formula determines the area of the resulting isosceles triangle, which is the maximum possible among all triangles with the same base and perimeter. On the other hand, if the area and perimeter are fixed, this formula can be used to recover the base length, but not uniquely: there are in general two distinct isosceles triangles with given area T and perimeter p. When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral.


Angle bisector length

If the two equal sides have length a and the other side has length b, then the internal
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
t from one of the two equal-angled vertices satisfies :\frac > t > \frac as well as :t<\frac; and conversely, if the latter condition holds, an isosceles triangle parametrized by a and t exists. The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by C. L. Lehmus. Its other namesake,
Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
, was one of the first to provide a solution. Although originally formulated only for internal angle bisectors, it works for many (but not all) cases when, instead, two external angle bisectors are equal. The 30-30-120 isosceles triangle makes a boundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external).


Radii

The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles. The radius of the inscribed circle of an isosceles triangle with side length a, base b, and height h is: :\frac. The center of the circle lies on the symmetry axis of the triangle, this distance above the base. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles. The radius of the
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
is: :\frac. The center of the circle lies on the symmetry axis of the triangle, this distance below the apex.


Inscribed square

For any isosceles triangle, there is a unique square with one side collinear with the base of the triangle and the opposite two corners on its sides. The Calabi triangle is a special isosceles triangle with the property that the other two inscribed squares, with sides collinear with the sides of the triangle, are of the same size as the base square. A much older theorem, preserved in the works of
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He ...
, states that, for an isosceles triangle with base b and height h, the side length of the inscribed square on the base of the triangle is :\frac.


Isosceles subdivision of other shapes

For any integer n \ge 4, any
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
can be partitioned into n isosceles triangles. In a
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
, the median from the hypotenuse (that is, the line segment from the midpoint of the hypotenuse to the right-angled vertex) divides the right triangle into two isosceles triangles. This is because the midpoint of the hypotenuse is the center of the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
of the right triangle, and each of the two triangles created by the partition has two equal radii as two of its sides. Similarly, an
acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
can be partitioned into three isosceles triangles by segments from its circumcenter, but this method does not work for obtuse triangles, because the circumcenter lies outside the triangle. Generalizing the partition of an acute triangle, any
cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
that contains the center of its circumscribed circle can be partitioned into isosceles triangles by the radii of this circle through its vertices. The fact that all radii of a circle have equal length implies that all of these triangles are isosceles. This partition can be used to derive a formula for the area of the polygon as a function of its side lengths, even for cyclic polygons that do not contain their circumcenters. This formula generalizes
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
for triangles and
Brahmagupta's formula In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version (Bretschneider's formula) can be used with non-cyclic ...
for
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
s. Either
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
of a
rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
divides it into two
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
isosceles triangles. Similarly, one of the two diagonals of a
kite A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
divides it into two isosceles triangles, which are not congruent except when the kite is a rhombus.


Applications


In architecture and design

Isosceles triangles commonly appear in
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
as the shapes of
gable A gable is the generally triangular portion of a wall between the edges of intersecting roof pitches. The shape of the gable and how it is detailed depends on the structural system used, which reflects climate, material availability, and aesth ...
s and
pediment Pediments are gables, usually of a triangular shape. Pediments are placed above the horizontal structure of the lintel, or entablature, if supported by columns. Pediments can contain an overdoor and are usually topped by hood moulds. A pedimen ...
s. In
ancient Greek architecture Ancient Greek architecture came from the Greek-speaking people (''Hellenic'' people) whose culture flourished on the Greek mainland, the Peloponnese, the Aegean Islands, and in colonies in Anatolia and Italy for a period from about 900 BC unti ...
and its later imitations, the obtuse isosceles triangle was used; in
Gothic architecture Gothic architecture (or pointed architecture) is an architectural style that was prevalent in Europe from the late 12th to the 16th century, during the High and Late Middle Ages, surviving into the 17th and 18th centuries in some areas. It e ...
this was replaced by the acute isosceles triangle. In the architecture of the Middle Ages, another isosceles triangle shape became popular: the Egyptian isosceles triangle. This is an isosceles triangle that is acute, but less so than the equilateral triangle; its height is proportional to 5/8 of its base. The Egyptian isosceles triangle was brought back into use in modern architecture by Dutch architect
Hendrik Petrus Berlage Hendrik Petrus Berlage (21 February 1856 – 12 August 1934) was a Dutch architect. He is considered one of the fathers of the architecture of the Amsterdam School. Life and work Hendrik Petrus Berlage, son of Nicolaas Willem Berlage and An ...
.
Warren truss Warren Errol Truss, (born 8 October 1948) is a former Australian politician who served as the 16th Deputy Prime Minister of Australia and Minister for Infrastructure and Regional Development in the Abbott Government and the Turnbull Governm ...
structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength. Surfaces
tessellated A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
by obtuse isosceles triangles can be used to form
deployable structure A deployable structure is a structure that can change shape so as to significantly change its size. Examples of deployable structures are umbrellas, some tensegrity structures, bistable structures, some Origami shapes and scissor-like structures. ...
s that have two stable states: an unfolded state in which the surface expands to a cylindrical column, and a folded state in which it folds into a more compact prism shape that can be more easily transported. The same tessellation pattern forms the basis of Yoshimura buckling, a pattern formed when cylindrical surfaces are axially compressed, and of the Schwarz lantern, an example used in mathematics to show that the area of a smooth surface cannot always be accurately approximated by polyhedra converging to the surface. In
graphic design Graphic design is a profession, academic discipline and applied art whose activity consists in projecting visual communications intended to transmit specific messages to social groups, with specific objectives. Graphic design is an interdiscipli ...
and the
decorative arts ] The decorative arts are arts or crafts whose object is the design and manufacture of objects that are both beautiful and functional. It includes most of the arts making objects for the interiors of buildings, and interior design, but not usual ...
, isosceles triangles have been a frequent design element in cultures around the world from at least the
Early Neolithic The Neolithic period, or New Stone Age, is an Old World archaeological period and the final division of the Stone Age. It saw the Neolithic Revolution, a wide-ranging set of developments that appear to have arisen independently in several parts ...
to modern times. They are a common design element in
flag A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
s and
heraldry Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, rank and pedigree. Armory, the best-known branch ...
, appearing prominently with a vertical base, for instance, in the
flag of Guyana The flag of Guyana, known as The Golden Arrowhead, has been the national flag of Guyana since May 1966 when the country became independent from the United Kingdom. It was designed by Whitney Smith, an American vexillologist (though originally ...
, or with a horizontal base in the
flag of Saint Lucia The flag of Saint Lucia consists of a cerulean blue field charged with a yellow triangle in front of a white-edged black isosceles triangle. Adopted in 1967 to replace the British Blue Ensign Defacement (flag), defaced with the arms of the colon ...
, where they form a stylized image of a mountain island. They also have been used in designs with religious or mystic significance, for instance in the Sri Yantra of Hindu meditational practice.


In other areas of mathematics

If a
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
with real coefficients has three roots that are not all
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, then when these roots are plotted in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
as an
Argand diagram In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
they form vertices of an isosceles triangle whose axis of symmetry coincides with the horizontal (real) axis. This is because the complex roots are
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s and hence are symmetric about the real axis. In
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
has been studied in the special case that the three bodies form an isosceles triangle, because assuming that the bodies are arranged in this way reduces the number of degrees of freedom of the system without reducing it to the solved
Lagrangian point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...
case when the bodies form an equilateral triangle. The first instances of the three-body problem shown to have unbounded oscillations were in the isosceles three-body problem.


History and fallacies

Long before isosceles triangles were studied by the
ancient Greek mathematicians Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
, the practitioners of Ancient Egyptian mathematics and
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babyl ...
knew how to calculate their area. Problems of this type are included in the
Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
and Rhind Mathematical Papyrus.. Although "many of the early Egyptologists" believed that the Egyptians used an inexact formula for the area, half the product of the base and side,
Vasily Vasilievich Struve Vasily Vasilievich Struve (russian: Василий Васильевич Струве) ( in Petersburg, Russian Empire – September 15, 1965 in Leningrad) was a Soviet orientalist from the Struve family, the founder of the Soviet scientific scho ...
championed the view that they used the correct formula, half the product of the base and height . This question rests on the translation of one of the words in the Rhind papyrus, and with this word translated as height (or more precisely as the ratio of height to base) the formula is correct .
The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid. This result has been called the ''
pons asinorum In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the ''pons asinorum'' (, ), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in E ...
'' (the bridge of asses) or the isosceles triangle theorem. Rival explanations for this name include the theory that it is because the diagram used by Euclid in his demonstration of the result resembles a bridge, or because this is the first difficult result in Euclid, and acts to separate those who can understand Euclid's geometry from those who cannot. A well-known
fallacy A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was intr ...
is the false proof of the statement that ''all triangles are isosceles''. Robin Wilson credits this argument to
Lewis Carroll Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ...
, who published it in 1899, but
W. W. Rouse Ball Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...
published it in 1892 and later wrote that Carroll obtained the argument from him. The fallacy is rooted in Euclid's lack of recognition of the concept of ''betweenness'' and the resulting ambiguity of ''inside'' versus ''outside'' of figures.


Notes


References

* * * * * * * * * * * * *. See in particular p. 111. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *. * * *


External links

* {{Polygons Types of triangles