Steiner-Lehmus theorem

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$|AE|=|BD|,\,\alpha=\beta,\, \gamma=\delta$ $\Rightarrow \triangle ABC \text{ is isosceles}$

The Steiner-Lehmus theorem is a theorem in elementary geometry, that was first formulated by C. L. Lehmus and then subsequently proved by Jakob Steiner.

Any triangle with two angle bisectors of equal lengths is isosceles.

The theorem was first mentioned in 1840 in a letter by C. L. Lehmus to C. Sturm, in which he asked for a purely geometric proof. C. Sturm passed the request on to other mathematicians and Jakob Steiner was one of the first that provided a solution. The theorem became a rather popular topic in elementary geometry ever since with a somewhat regular publication of articles on it.^[1] ^[2]

[edit] Impossibility of a Direct Proof

The Steiner-Lehmus theorem can be proved using elementary geometry by proving the contrapositive statement. There is some controversy over whether a "direct" proof is possible; allegedly "direct" proofs have been published, but not everyone agrees that these proofs are "direct." John Conway^[3] has argued that there can be no "equality-chasing" proof because the theorem is false over an arbitrary field. However, until someone formulates a precise definition of what a "direct proof" is, there remains room for debate.