The authors consider a triangular interpolation scheme on a continuous piecewise \(C^1\) curve of the complex plane and denote the closure of this triangular scheme by \(\Gamma\). Given a meromorphic function \(f\) with no singularity on \(\Gamma\) they examine the region of convergence of the sequence of interpolating polynomials to the function \(f\). In particular they show that the sequence of interpolating polynomials \(\{P_n\}_n\) is divergent on all points of \(\Gamma_{\text{out}}\) except on a set of measure zero, where \(\Gamma_{\text{out}}\) denotes the subset of \(\Gamma\) outside of the convergence region.