John Horton Conway lived to discover the Mathematics behind problems, always working to isolate a pure, essential kernel of truth. He loved to communicate these simple truths to others, often changing the way they thought. Everything John touched turned to Mathematics, and to very beautiful Mathematics. John was generous with his mathematical riches; he gave them to everyone that showed interest — whether at the coffee house, at the sushi restaurant, or in Mathematics departments. He was a magnet for all mathematicians, and he welcomed all who came to him. John would find a way to start with some simple calculation, a game or puzzle and turn it into whatever he wanted to explain. John often chose problems about games and recreational topics, but the insights he derived changed our understanding of several very serious branches of Mathematics.

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.

This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

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.