Steiner-Lehmus Theorem/Proof 4

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

Lemma $1$

In the same circle, let one chord $PR$, be larger than another, $PQ$.

Then the inscribed angle subtended by $PR$, with its vertex in the major arc, is larger.

$\Box$

Lemma $2$

Let $\triangle ABC$ be a triangle.

Let $\angle ABC$ be bisected by $BM$.

Let $\angle ACB$ be bisected by $CN$.

Let $\angle ABC < \angle ACB$.

Then:

$CN < BM$

$\Box$

Let $\triangle ABC$ be a triangle.

Let $\angle ABC$ be bisected by $BM$

Let $\angle ACB$ be bisected by $CN$.

Let $BM = CN$.

Suppose $\angle ABC < \angle ACB$.

By Lemma $2$:

$CN < BM$

This is a contradiction.

$\Box$

Suppose $\angle ABC < \angle ACB$.

By Lemma $2$:

$CN > BM$

This is a contradiction.

$\Box$

Since $\angle ABC$ can be neither less than nor greater than $\angle ACB$, $\angle ABC = \angle ACB$.

By Triangle with Two Equal Angles is Isosceles:

$\triangle ABC$ is an isosceles triangle.

The result follows.

$\blacksquare$

Sources

• 1967: H.S.M. Coxeter and S.L. Greitzer: Geometry Revisited: Chapter $1$ : "Points and Lines Connected with a Triangle."