The Steiner-Lehmus angle-bisector theorem

In 1840 C. L. Lehmus sent the following problem to Charles Sturm: ‘If two angle bisectors of a triangle have equal length, is the triangle necessarily isosceles?’ The answer is ‘yes’, and indeed we have the reverse-comparison theorem : Of two unequal angles, the larger has the shorter bisector (see [1, 2]). Sturm passed the problem on to other mathematicians, in particular to the great Swiss geometer Jakob Steiner, who provided a proof. In this paper we give several proofs and discuss the old query: ‘Is there a direct proof?’ before suggesting that this is no longer the right question to ask. We go on to discuss all cases when an angle bisector (internal orexternal) of some angle is equal to one of another.