By resumming the Feynman graphs which contribute to any gauge-invariant process we explicitly construct, at one-loop order, a three-gluon vertex for QCD which is completely independent of the choice of gauge. This vertex satisfies a Ward identity of the type encountered in ghost-free gauges, relating the vertex to the proper self-energy of a previously constructed gluon propagator, also found by resumming graphs; like the vertex, this self-energy is completely gauge invariant. We also derive the gauge-invariant propagator and vertex via a second related technique which minimizes the dependence on embedding these objects in a gauge-invariant process; the same results are found as in the first technique. These results motivate a toy model of the nonlinear Schwinger-Dyson equation satisfied by the exact gauge-invariant three-gluon vertex. This model is nonperturbative and has infrared singularities, which we can remove via gluon mass generation; it shows many interesting features expected of QCD, such as a beta function which is not Borel summable in perturbation theory.