Editing Isosceles triangle (section)

===Angle bisector length===
If the two equal sides have length <math>a</math> and the other side has length <math>b</math>, then the internal [[angle bisector]] <math>t</math> from one of the two equal-angled vertices satisfies{{sfnp|Arslanagić}}
:<math>\frac{2ab}{a+b} > t > \frac{ab\sqrt{2}}{a+b}</math>
as well as
:<math>t<\frac{4a}{3};</math>
and conversely, if the latter condition holds, an isosceles triangle parametrized by <math>a</math> and <math>t</math> exists.{{sfnp|Oxman|2005}}

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]], was one of the first to provide a solution.{{sfnp|Gilbert|MacDonnell|1963}}
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).{{sfnp|Conway|Ryba|2014}}