A Short Trigonometric Proof of the Steiner-Lehmus Theorem

We give a short trigonometric proof of the Steiner-Lehmus theorem. The well known Steiner-Lehmus theorem states that if the internal angle bisectors of two angles of a triangle are equal, then the triangle is isosceles. Unlike its trivial converse, this challenging statement has attracted a lot of attention since 1840, when Professor Lehmus of Berlin wrote to Sturm asking for a purely geometrical proof. Proofs by Rougevain, Steiner, and Lehmus himself appeared in the following few years. Since then, a great number of people, including several renowned mathematicians, took interest in the problem, resulting in as many as 80 different proofs. Extensive histories are given in [14], [15], [16], and [21], and biographies and lists of references can be found in [33], [37], and [19]. More references will be referred to later when we discuss generalizations and variations of the theorem. In this note, we present a new trigonometric proof of the theorem. Compared with the existing proofs, such as the one given in [17, pp. 194–196], it is also short and simple. It runs as follows.