Editing Isosceles triangle (section)

==History and fallacies==
Long before isosceles triangles were studied by the [[Greek mathematics|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]].<ref>{{harvtxt|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 {{harv|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 {{harv|Gunn|Peet|1929|pages=173–174}}.</ref>

The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid.{{sfnp|Heath|1926|loc=[https://archive.org/details/bwb_S0-AHZ-704_1/page/251/ I.5, p. 251]}} 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.{{sfnp|Venema|2006|loc=p. 89}}

A well-known [[Mathematical fallacy#Geometry|fallacy]] is the false proof of the statement that ''all triangles are isosceles'', first published by [[W. W. Rouse Ball]] in 1892,{{sfnp|Ball|Coxeter|1987}} and later republished in [[Lewis Carroll]]'s posthumous ''Lewis Carroll Picture Book''.<ref>{{harvtxt| Carroll|1899}}. See also {{harvtxt|Wilson|2008}}.</ref> 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.{{sfnp|Specht|Jones|Calkins|Rhoads|2015}}