This work develops and tests a trust region algorithm for the nonlinear equality constrained optimization problem. Our goal is to develop a robust algorithm that can handle lack of second-order sufficiency away from the solution in a natural way. Celis, Dennis and Tapia [1985] give a trust region algorithm for this problem, but in certain situations their trust region subproblem is too difficult to solve. The algorithm given here is based on the restriction of the trust region subproblem given by Celis, Dennis and Tapia [1985] to a relevant two dimensional subspace. This restriction greatly facilitates the solution of the subproblem. The trust region subproblem that is the focus of this work requires the minimization of a possibly non-convex quadratic subject to two quadratic constraints in two dimensions. The solution of this problem requires the determination of all the global solutions, and the non-global solution, if it exists, to the standard unconstrained trust region subproblem. Algorithms for approximating a single global solution to the unconstrained trust region subproblem have been well-established. Analytical expressions for all of the solutions will be derived for a number of special cases, and necessary and sufficient conditions are given for the existence of a non-global solution for the general case of the two-dimensional unconstrained trust region subproblem. Finally, numerical results are presented for a preliminary implementation of the nonlinear programming algorithm, and these results verify that it is indeed robust.