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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 edges and three 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, any three points, when non- colli ...
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 as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. 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 pedim ...
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 aest ...
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 of its base. The two angles opposite the legs are equal and are always acute, 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, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
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 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 ...
(with three equal sides) a special case of isosceles triangles. A triangle that is not isosceles (having three unequal sides) is called
scalene Scalene may refer to: * A scalene triangle, one in which all sides and angles are not the same. * A scalene ellipsoid, one in which the lengths of all three semi-principal axes are different * Scalene muscles of the neck * Scalene tubercle The sc ...
. "Isosceles" is made from the Greek roots "isos" (equal) and "skelos" (leg). The same word is used, for instance, for isosceles trapezoids, trapezoids with two equal sides, and for isosceles sets, 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 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. 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 mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, 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'', 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 status, socioeconomic factors like wealth, income, Race (human categorization), race, education, ethnicity, gender, Job, occupation, socia ...
: 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 colo ...
, with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles. As well as the isosceles right triangle, several other specific shapes of isosceles triangles have been studied. These include the
Calabi triangle The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an obtuse isosceles triangle with an irrational but algebraic ratio ...
(a triangle with three congruent inscribed squares), the golden triangle 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 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 solids, the triakis tetrahedron, triakis octahedron, tetrakis hexahedron, pentakis dodecahedron, and triakis icosahedron, 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, quadrila ...
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 from the apex to the base, *the median from the apex to the midpoint of the base, *the perpendicular bisector of the base within the triangle, *the segment within the triangle of the unique axis of symmetry of the triangle, and *the segment within the triangle of the Euler line of the triangle, except when the triangle is equilateral. 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 opposit ...
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 (the intersection of its three altitudes), its centroid (the intersection of its three medians), and its circumcenter (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 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 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 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 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 The Steiner–Lehmus theorem, a theorem in elementary geometry, was formulated by C. L. Lehmus and subsequently proved by Jakob Steiner. It states: : ''Every triangle with two angle bisectors of equal lengths is isosceles''. The theorem was ...
states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by
C. L. Lehmus Daniel Christian Ludolph Lehmus (July 3, 1780 in Soest – January 18, 1863 in Berlin) was a German mathematician, who today is best remembered for the Steiner–Lehmus theorem, that was named after him. Lehmus was the grandson of the German poe ...
. Its other namesake, Jakob Steiner, 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 An edge case is a problem or situation that occurs only at an extreme (maximum or minimum) operating parameter. For example, a stereo speaker might noticeably distort audio when played at maximum volume, even in the absence of any other extreme ...
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 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 The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an obtuse isosceles triangle with an irrational but algebraic ratio ...
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 edges and three 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, any three points, when non- colli ...
can be partitioned into n isosceles triangles. In a right triangle, 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 pol ...
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 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 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 for triangles and Brahmagupta's formula for cyclic quadrilaterals. Either diagonal 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. Th ...
divides it into two congruent 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 buildings ...
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 aest ...
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 pedim ...
s. In ancient Greek architecture 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. I ...
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. Warren truss structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength. Surfaces tessellated by obtuse isosceles triangles can be used to form deployable structures 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 and the decorative arts, isosceles triangles have been a frequent design element in cultures around the world from at least the Early Neolithic to modern times. They are a common design element in flags and heraldry, 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 origina ...
, or with a horizontal base in the flag of Saint Lucia, 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 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, then when these roots are plotted in the complex plane as an Argand diagram 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 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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
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 ...
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, the practitioners of Ancient Egyptian mathematics and Babylonian mathematics knew how to calculate their area. Problems of this type are included in the Moscow Mathematical Papyrus 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 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'' (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 int ...
is the false proof of the statement that ''all triangles are isosceles''. Robin Wilson credits this argument to Lewis Carroll, 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