A triangle group has a presentation of the form,

$G=\langle a, b; a^{\alpha}, b^{\beta}, c^{\gamma}, abc\rangle, \alpha, \beta, \gamma \geq 2$

(I believe that these are also called von Dyke groups, or "ordinary" triangle groups, with triangle groups being something slightly different, but names are beside the point). I have been reading the Fine and Rosenberg paper which proves that these groups are conjugacy separable ("Conjugacy separability of Fuchsian groups and related questions"; the proof, and the statement below, can also be found in their book, "Algebraic generalizations of discrete groups"), and in it the authors state,

"The conjugacy classes of elements of finite order...are given by the conjugacy classes {$\langle a\rangle$}, {$\langle b\rangle$}, {$\langle c\rangle$}".

This statement is given without proof or reference. I was therefore wondering if someone could provide either a proof or a reference for this?

I understand where the comment comes from - it is the obvious generalisation of the one-relator groups case (here we are dealing with one-relator products, which generalise one-relator groups). However, I cannot seem to find a proof of the statement in the literature, although I am sure it must be there. Unless, of course, I am simply missing something and the result is obvious...