  We have studied the $\gamma p \to p \pi^0 \eta$ reaction paying attention to
the two main mechanisms at low energies, the $\gamma p \to \Delta(1700) \to
\eta \Delta(1232)$ and the $\gamma p \to \Delta(1700) \to \pi N(1535)$. Both of
them are driven by the photoexcitation of the $\Delta(1700)$ and the second one
involves a mechanism that leads to a triangle singularity. We are able to
evaluate quantitatively the cross section for this process and show that it
agrees with the experimental determination. Yet, there are some differences
with the standard partial wave analysis which does not include explicitly the
triangle singularity. The exercise also shows the convenience to explore
possible triangle singularities in other reactions and how a standard partial
analysis can be extended to accommodate them.
