Editing Isosceles triangle (section)

==Terminology, classification, and examples==
[[Euclid]] defined an isosceles triangle as a triangle with exactly two equal sides,{{sfnp|Heath|1926|loc=[https://archive.org/details/bwb_S0-AHZ-704_1/page/187/ Def. 20, p. 187]}} 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.{{sfnp|Stahl|2003|loc=[https://books.google.com/books?id=jLk7lu3bA1wC&pg=PA37 p. 37]}} A triangle that is not isosceles (having three unequal sides) is called [[scalene triangle|scalene]].{{sfnp|Usiskin|Griffin|2008|page=4}}
"Isosceles" is made from the [[List of Greek and Latin roots in English|Greek roots]] "isos" (equal) and "skelos" (leg). The same word is used, for instance, for [[isosceles trapezoid]]s, trapezoids with two equal sides,{{sfnp|Usiskin|Griffin|2008|page=41}} and for [[isosceles set]]s, sets of points every three of which form an isosceles triangle.{{sfnp|Ionin|2009}}

In an isosceles triangle that has exactly two equal sides, the equal sides are called [[Catheti|legs]] and the third side is called the [[base (geometry)|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''.{{sfnp|Jacobs|1974|page=144}} The vertex opposite the base is called the [[apex (geometry)|apex]].{{sfnp|Gottschau|Haverkort|Matzke|2018}} In the equilateral triangle case, since all sides are equal, any side can be called the base.{{sfnp|Lardner|1840|page=46}}

{{multiple image
|perrow = 2
|total_width=480
|header=Special isosceles triangles
|image1=45-45-triangle.svg
|caption1=[[Isosceles right triangle]]
|image2=Calabi triangle.svg
|caption2=Three congruent inscribed squares in the [[Calabi triangle]]
|image3=Golden triangle (math).svg
|caption3=A [[Golden triangle (mathematics)|golden triangle]] subdivided into a smaller golden triangle and golden gnomon
|image4=1-uniform 4 dual.svg
|caption4=The [[triakis triangular tiling]]
}}
{{multiple image
|total_width=600
|header=Catalan solids with isosceles triangle faces
|image1=Triakistetrahedron.jpg
|caption1=[[Triakis tetrahedron]]
|image2=Triakisoctahedron.jpg
|caption2=[[Triakis octahedron]]
|image3=Tetrakishexahedron.jpg
|caption3=[[Tetrakis hexahedron]]
|image4=Pentakisdodecahedron.jpg
|caption4=[[Pentakis dodecahedron]]
|image5=Triakisicosahedron.jpg
|caption5=[[Triakis icosahedron]]
}}

Whether an isosceles triangle is [[Acute and obtuse triangles|acute, right or obtuse]] depends only on the angle at its apex. In [[Euclidean 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.{{sfnp|Lardner|1840|page=46}} 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.{{sfnp|Gottschau|Haverkort|Matzke|2018}} In [[Edwin Abbott Abbott|Edwin Abbott]]'s book ''[[Flatland]]'', this classification of shapes was used as a satire of [[social hierarchy]]: isosceles triangles represented the [[working class]], with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles.{{sfnp|Barnes|2012}}

As well as the [[isosceles right triangle]], several other specific shapes of isosceles triangles have been studied.
These include the [[Calabi triangle]] (a triangle with three congruent inscribed squares),{{sfnp|Conway|Guy|1996}} the [[Golden triangle (mathematics)|golden triangle]] and [[golden gnomon]] (two isosceles triangles whose sides and base are in the [[golden ratio]]),{{sfnp|Loeb|1992}} the 80-80-20 triangle appearing in the [[Langley's Adventitious Angles]] puzzle,{{sfnp|Langley|1922}} and the 30-30-120 triangle of the [[triakis triangular tiling]].
Five [[Catalan solid]]s, the [[triakis tetrahedron]], [[triakis octahedron]], [[tetrakis hexahedron]], [[pentakis dodecahedron]], and [[triakis icosahedron]], each have isosceles-triangle faces, as do infinitely many [[Pyramid (geometry)|pyramid]]s{{sfnp|Lardner|1840|page=46}} and [[bipyramid]]s.{{sfnp|Montroll|2009}}