  The decay of $\Lambda_c^+$ into $\pi^+\pi^0\Lambda(1405)$ with the
$\Lambda(1405)$ decay into $\pi^0\Sigma^0$ through a triangle diagram is
studied. This process is initiated by $ \Lambda_c^+ \to \pi^+ \bar{K}^*N $,
then the $\bar{K}^*$ decays into $\bar{K} \pi$ and $\bar{K} N$ produce the
$\Lambda(1405)$ through a triangle loop containing $\bar{K}^*N\bar{K}$ which
develops a singularity around $1890$ MeV. This process is prohibited by the
isospin symmetry, but the decay into this channel is enhanced by the
contribution of the triangle diagram, which is sensitive to the mass of the
internal particles. We find a narrow peak in the $\pi^0\Sigma^0$ invariant mass
distribution, which originates from the $\Lambda(1405)$ amplitude, but is tied
to the mass differences between the charged and neutral $\bar{K}$ or $N$
states. The observation of the unavoidable peak of the triangle singularity in
the isospin-violating $\Lambda(1405)$ production would provide further support
for hadronic molecular picture of the $\Lambda(1405)$ and further information
on the $\bar{K} N $ interaction.
