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Isosceles triangle

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Triangle with at least two sides congruent

"Isosceles" redirects here. For other uses, seeIsosceles (disambiguation).

Isosceles triangle
Isosceles triangle
Type	triangle
Edges andvertices	3
Schläfli symbol	( ) ∨ { }
Symmetry group	Dih₂, [ ], (*), order 2
Properties	convex,cyclic
Dual polygon	Self-dual

Ingeometry, anisosceles triangle (/aɪˈsɒsəliːz/) is atriangle that has twosides of equal length and twoangles of equal measure. Sometimes it is specified as havingexactly two sides of equal length, and sometimes as havingat least two sides of equal length, the latter version thus including theequilateral triangle as aspecial case.Examples of isosceles triangles include theisosceles right triangle, thegolden triangle, and the faces ofbipyramids and certainCatalan solids.

The mathematical study of isosceles triangles dates back toancient Egyptian mathematics andBabylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in thepediments andgables of buildings.

The two equal sides are called thelegs and the third side is called thebase 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 hasreflection symmetry across theperpendicular bisector of its base, which passes through the oppositevertex and divides the triangle into a pair ofcongruent right triangles. The two equal angles at the base (opposite the legs) are alwaysacute, so the classification of the triangle as acute, right, orobtuse depends only on the angle between its two legs.

Terminology, classification, and examples

Euclid defined an isosceles triangle as a triangle with exactly two equal sides,^[1] 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 makesequilateral triangles (with three equal sides) a special case of isosceles triangles.^[2] A triangle that is not isosceles (having three unequal sides) is calledscalene.^[3]"Isosceles" is made from theGreek roots "isos" (equal) and "skelos" (leg). The same word is used, for instance, forisosceles trapezoids, trapezoids with two equal sides,^[4] and forisosceles sets, sets of points every three of which form an isosceles triangle.^[5]

In an isosceles triangle that has exactly two equal sides, the equal sides are calledlegs and the third side is called thebase. The angle included by the legs is called thevertex angle and the angles that have the base as one of their sides are called thebase angles.^[6] The vertex opposite the base is called theapex.^[7] In the equilateral triangle case, since all sides are equal, any side can be called the base.^[8]

Special isosceles triangles

Isosceles right triangle

Three congruent inscribed squares in theCalabi triangle

Agolden triangle subdivided into a smaller golden triangle and golden gnomon

Thetriakis triangular tiling

Catalan solids with isosceles triangle faces

Triakis tetrahedron

Triakis octahedron

Tetrakis hexahedron

Pentakis dodecahedron

Triakis icosahedron

Whether an isosceles triangle isacute, right or obtuse depends only on the angle at its apex. InEuclidean geometry, 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.^[8] 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.^[7] InEdwin Abbott's bookFlatland, this classification of shapes was used as a satire ofsocial hierarchy: isosceles triangles represented theworking class, with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles.^[9]

As well as theisosceles right triangle, several other specific shapes of isosceles triangles have been studied.These include theCalabi triangle (a triangle with three congruent inscribed squares),^[10] thegolden triangle andgolden gnomon (two isosceles triangles whose sides and base are in thegolden ratio),^[11] the 80-80-20 triangle appearing in theLangley's Adventitious Angles puzzle,^[12] and the 30-30-120 triangle of thetriakis triangular tiling.FiveCatalan solids, thetriakis tetrahedron,triakis octahedron,tetrakis hexahedron,pentakis dodecahedron, andtriakis icosahedron, each have isosceles-triangle faces, as do infinitely manypyramids^[8] andbipyramids.^[13]

Formulas

Height

For any isosceles triangle, the following sixline segments coincide:

thealtitude, a line segment from the apex perpendicular to the base,^[14]
theangle bisector from the apex to the base,^[14]
themedian from the apex to the midpoint of the base,^[14]
theperpendicular bisector of the base within the triangle,^[14]
the segment within the triangle of the uniqueaxis of symmetry of the triangle, and^[14]
the segment within the triangle of theEuler line of the triangle, except when the triangle isequilateral.^[15]

Their common length is the height $h {\displaystyle h}$ of the triangle.If the triangle has equal sides of length $a {\displaystyle a}$ and base of length $b {\displaystyle b}$ ,thegeneral triangle formulas for the lengths of these segments all simplify to^[16]

h={\sqrt {a^{2}-{\frac {b^{2}}{4}}}}.

This formula can also be derived from thePythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles.^[17]

The Euler line of any triangle goes through the triangle'sorthocenter (the intersection of its three altitudes), itscentroid (the intersection of its three medians), and itscircumcenter (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. Theincenter of the triangle also lies on the Euler line, something that is not true for other triangles.^[15] If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles.^[18]

Area

The area $T {\displaystyle T}$ 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:^[16]

T={\frac {b}{4}}{\sqrt {4a^{2}-b^{2}}}.

The same area formula can also be derived fromHeron's formula for the area of a triangle from its three sides. However, applying Heron's formula directly can benumerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between thesemiperimeter and side length in those triangles.^[19]

If the apex angle $(\theta )$ and leg lengths $(a)$ of an isosceles triangle are known, then the area of that triangle is:^[20]

T={\frac {1}{2}}a^{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.^[21]

Perimeter

The perimeter $p {\displaystyle p}$ of an isosceles triangle with equal sides $a {\displaystyle a}$ and base $b {\displaystyle b}$ is just^[16]

p=2a+b.

As in any triangle, the area $T {\displaystyle T}$ and perimeter $p {\displaystyle p}$ are related by theisoperimetric inequality^[22]

p^{2}>12{\sqrt {3}}T.

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^[23]

2pb^{3}-p^{2}b^{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.^[24]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 $T {\displaystyle T}$ and perimeter $p {\displaystyle p}$ . When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral.^[25]

Angle bisector length

If the two equal sides have length $a {\displaystyle a}$ and the other side has length $b {\displaystyle b}$ , then the internalangle bisector $t {\displaystyle t}$ from one of the two equal-angled vertices satisfies^[26]

{\frac {2ab}{a+b}}>t>{\frac {ab{\sqrt {2}}}{a+b}}

as well as

t<{\frac {4a}{3}};

and conversely, if the latter condition holds, an isosceles triangle parametrized by $a {\displaystyle a}$ and $t {\displaystyle t}$ exists.^[27]

TheSteiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 byC. L. Lehmus. Its other namesake,Jakob Steiner, was one of the first to provide a solution.^[28]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 aboundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external).^[29]

Radii

Isosceles triangle showing its circumcenter (blue), centroid (red), incenter (green), and symmetry axis (purple)

The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles.^[30]The radius of theinscribed circle of an isosceles triangle with side length $a {\displaystyle a}$ , base $b {\displaystyle b}$ , and height $h {\displaystyle h}$ is:^[16]

{\frac {2ab-b^{2}}{4h}}.

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.^[31]

The radius of thecircumscribed circle is:^[16]

{\frac {a^{2}}{2h}}.

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. TheCalabi 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.^[10] A much older theorem, preserved in the works ofHero of Alexandria,states that, for an isosceles triangle with base $b {\displaystyle b}$ and height $h {\displaystyle h}$ , the side length of the inscribed square on the base of the triangle is^[32]

{\frac {bh}{b+h}}.

Isosceles subdivision of other shapes

Partition of acyclic pentagon into isosceles triangles by radii of its circumcircle

For any integer $n\geq 4$ , anytriangle can be partitioned into $n {\displaystyle n}$ isosceles triangles.^[33]In aright 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 thecircumcircle of the right triangle, and each of the two triangles created by the partition has two equal radii as two of its sides.^[34]Similarly, anacute triangle can be partitioned into three isosceles triangles by segments from its circumcenter,^[35] but this method does not work for obtuse triangles, because the circumcenter lies outside the triangle.^[30]

Generalizing the partition of an acute triangle, anycyclic 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 generalizesHeron's formula for triangles andBrahmagupta's formula forcyclic quadrilaterals.^[36]

Eitherdiagonal of arhombus divides it into twocongruent isosceles triangles. Similarly, one of the two diagonals ofakite divides it into two isosceles triangles, which are not congruent except when the kite is a rhombus.^[37]

Applications

In architecture and design

Obtuse isosceles pediment of thePantheon, Rome

Acute isosceles gable over the Saint-Etienne portal,Notre-Dame de Paris

Isosceles triangles commonly appear inarchitecture as the shapes ofgables andpediments. Inancient Greek architecture and its later imitations, the obtuse isosceles triangle was used; inGothic architecture this was replaced by the acute isosceles triangle.^[8]

In thearchitecture 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.^[38] The Egyptian isosceles triangle was brought back into use in modern architecture by Dutch architectHendrik Petrus Berlage.^[39]

Detailed view of a modifiedWarren truss with verticals

Warren truss structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength.^[40]Surfacestessellated by obtuse isosceles triangles can be used to formdeployable 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.^[41] The same tessellation pattern forms the basis ofYoshimura buckling, a pattern formed when cylindrical surfaces are axially compressed,^[42] and of theSchwarz 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.^[43]

Flag of Saint Lucia

Ingraphic design and thedecorative arts, isosceles triangles have been a frequent design element in cultures around the world from at least theEarly Neolithic^[44] to modern times.^[45] They are a common design element inflags andheraldry, appearing prominently with a vertical base, for instance, in theflag of Guyana, or with a horizontal base in theflag of Saint Lucia, where they form a stylized image of a mountain island.^[46]

They also have been used in designs with religious or mystic significance, for instance in theSri Yantra ofHindu meditational practice.^[47]

In other areas of mathematics

If acubic equation with real coefficients has three roots that are not allreal numbers, then when these roots are plotted in thecomplex plane as anArgand diagram they form vertices of an isosceles triangle whose axis of symmetry coincides with the horizontal (real) axis. This is because the complex roots arecomplex conjugates and hence are symmetric about the real axis.^[48]

Incelestial mechanics, thethree-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 ofdegrees of freedom of the system without reducing it to the solvedLagrangian point 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.^[49]

History and fallacies

Long before isosceles triangles were studied by theancient Greek mathematicians, the practitioners ofAncient Egyptian mathematics andBabylonian mathematics knew how to calculate their area. Problems of this type are included in theMoscow Mathematical Papyrus andRhind Mathematical Papyrus.^[50]

The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid.^[51] This result has been called thepons 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.^[52]

A well-knownfallacy is the false proof of the statement thatall triangles are isosceles, first published byW. W. Rouse Ball in 1892,^[53] and later republished inLewis Carroll's posthumousLewis Carroll Picture Book.^[54] The fallacy is rooted in Euclid's lack of recognition of the concept ofbetweenness and the resulting ambiguity ofinside versusoutside of figures.^[55]

Notes

^Heath (1926),Def. 20, p. 187.
^Stahl (2003),p. 37.
^Usiskin & Griffin (2008), p. 4.
^Usiskin & Griffin (2008), p. 41.
^Ionin (2009).
^Jacobs (1974), p. 144.
^^a ^bGottschau, Haverkort & Matzke (2018).
^^a ^b ^c ^dLardner (1840), p. 46.
^Barnes (2012).
^^a ^bConway & Guy (1996).
^Loeb (1992).
^Langley (1922).
^Montroll (2009).
^^a ^b ^c ^d ^eHadamard (2008), p. 23.
^^a ^bGuinand (1984).
^^a ^b ^c ^d ^eHarris & Stöcker (1998), p. 78.
^Salvadori & Wright (1998).
^Hadamard (2008), Exercise 5, p. 29.
^Kahan (2014).
^Young (2011), p. 298.
^Young (2011), p. 398.
^Alsina & Nelsen (2009), p. 71.
^Baloglou & Helfgott (2008), Equation (1).
^Wickelgren (2012).
^Baloglou & Helfgott (2008), Theorem 2.
^Arslanagić.
^Oxman (2005).
^Gilbert & MacDonnell (1963).
^Conway & Ryba (2014).
^^a ^bHarris & Stöcker (1998), p. 75.
^Alsina & Nelsen (2009), p. 67.
^Gandz (1940).
^Lord (1982). See alsoHadamard (2008, Exercise 340, p. 270).
^Posamentier & Lehmann (2012), p. 24.
^Bezdek & Bisztriczky (2015).
^Robbins (1995).
^Usiskin & Griffin (2008), p. 51.
^Lavedan (1947).
^Padovan (2002).
^Ketchum (1920).
^Pellegrino (2002).
^Yoshimura (1955).
^Schwarz (1890).
^Washburn (1984).
^Jakway (1922).
^Smith (2014).
^Bolton, Nicol & Macleod (1977).
^Bardell (2016).
^Diacu & Holmes (1999).
^Høyrup (2008). 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 (Clagett 1989).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 (Gunn & Peet 1929, pp. 173–174).
^Heath (1926),I.5, p. 251.
^Venema (2006), p. 89.
^Ball & Coxeter (1987).
^Carroll (1899). See alsoWilson (2008).
^Specht et al. (2015).

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External links

Weisstein, Eric W.,"Isosceles triangle",MathWorld{{cite web}}: CS1 maint: overridden setting (link)

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1–10 sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10)
11–20 sides	Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20)
>20 sides	Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞)

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