Following the work of Gonzalez-Vega, this paper is devoted to show how to use recent algorithmic tools of computational real algebraic geometry to solve the Birkhoff Interpolation Problem. We recall and partly improve two algorithms to find at least one point in each connected component of a real algebraic set defined by a single equation or a polynomial system of equations, both based on the computation of the critical points of a distance function. These algorithms are used to solve the Birkhoff Interpolation Problem in a case which was known to be an open problem.