In many applications, the discretization of continuous ill-posed inverse problems results in discrete ill-posed problems whose solution requires the use of regularization strategies. The L-curve criterium is a popular tool for choosing good regularized solutions, when the data noise norm is not a priori known. In this work, we propose replacing the original ill-posed inverse problem with a noise-independent equality constrained one and solving the corresponding first-order equations by the Newton method. The sequence of the computed iterates defines a new discrete L-curve. By numerical results, we show that good regularized solutions correspond with the corner of this L-curve.