Steiner-Lehmus Theorem

Theorem
Let $ABC$ be a triangle.

Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.

Let $\omega_\alpha = \omega_\beta$.

Then $ABC$ is an isosceles triangle.

Proof
Let $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.

By Length of Angle Bisector, $\omega_\alpha, \omega_\beta$ are given by:


 * $\omega_\alpha^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$


 * $\omega_\beta^2 = \dfrac {b a} {\paren {b + a}^2} \paren {\paren {b + a}^2 - c^2}$

Equating the two equations,

Therefore $ABC$ is an isosceles triangle.

Remarks
The converse of the theorem is also true, and is much easier to prove:

Let $ABC$ be an isosceles triangle with $BA = BC$.

Let $D$ be the point of intersection of the angle bisectors through the vertices $A$ with the side $BC$.

Likewise, let $E$ be the points of intersection of the angle bisectors through the vertices $C$ with the side $BA$.

Then we have: