  In this work, we study the role of triangle singularity in the $J/\psi \to
\gamma \bar{p} \Delta$ decay. We find that through a triangle mechanism,
involving a triangle loop composed by $\omega$, $\pi$ and $p$, this decay may
develop a triangle singularity and produce a visible peak in the invariant mass
$M_{\gamma\Delta}$ around 1.73 GeV with a width of 0.02 GeV. Such a triangle
mechanism may also cause significant spin effects on the final $\Delta$, which
can be detected by measuring its spin density matrix elements. Our calculations
show that the branching ratios due to the triangle mechanism is Br($J/\psi\to
\gamma \bar p\Delta,\Delta\to \pi p$)=$1.058\times 10^{-6}$. Hopefully, this
reaction can be investigated at BESIII and future experiments, e.g. Super
Tau-Charm Facility, and the narrow width of the induced structure, the moving
TS position and the distinct features of the spin density matrix elements of
the $\Delta$ may serve as signals for the triangle singularity mechanism.
