We consider the minimization of a convex function on a compact polyhedron defined by linear equality constraints and nonnegative variables. We define the Levenberg-Marquardt (L-M) and central trajectories starting at the analytic center and using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of the parameter. Based on this we develop an algorithm that starts computing primal-dual feasible points on the L-M trajectory and eventually moves to the central path. Our main theorem is particularly relevant in quadratic programming, where points on the primal-dual L-M trajectory can be calculated by means of a system of linear equations. We present some computational tests related to box constrained trust region subproblems.