  We present a novel barycentric interpolation algorithm designed for analytic
functions $f\in\mathcal{A}(E)$ defined on the complex plane. The algorithm,
which encompasses both polynomial and rational interpolation, is tailored to
handle singularities near $E$. Our method is applicable to regions $E$ bounded
by piecewise smooth Jordan curves, and it imposes no connectivity restrictions
on the region. The key feature of our approach lies in efficiently computing
discrete points via the numerical solution of Symm's integral equation,
enabling the construction of polynomial or rational barycentric interpolants.
Furthermore, our method provides relevant parameters for the equilibrium
potential, such as Robin's constant, which can be used to estimate convergence
rates. Numerical experiments demonstrate the convergence rate achieved by our
method in comparison to the theoretical convergence rate.
