Steiner-Lehmus Theorem/Proof 1

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

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


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


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

Equating $\omega_\alpha^2$ with $\omega_\beta^2$:

Therefore $ABC$ is an isosceles triangle.