Fields of definition for triangle groups as Fuchsian groups

The compact hyperbolic triangle group $\mathrm{\Delta }(p,q,r)$ admits a canonical representation to ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is $$K=\mathbf{Q}(\cos (\pi \mathrm{/}p),\cos (\pi \mathrm{/}q),\cos (\pi \mathrm{/}r))\mathrm{.}$$ We prove that there are exactly eleven such groups which are conjugate to subgroups of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(K)$. Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(L)$ for any totally real field $L$. This answers a question first raised by Waterman and Machlachlan, and also resolves (in the positive) five (interrelated) recent conjectures of McMullen.