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.


 * Steiner-Lehmus.png

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