Tethers are nanocylinders of lipid bilayer membrane, arising in situations ranging from micromanipulation experiments on synthetic vesicles to the formation of dynamic tubular networks in the Golgi apparatus. Relying on the extensive theoretical and experimental works aimed to understand the physics of individual tethers formation, we addressed the problem of the interaction between two nanotubes. By using a combination of micropipette manipulation and optical tweezers, we quantitatively studied the process of coalescence that occurred when the separation distance between both vesicle-tether junctions became smaller than a threshold length. Our experiments, which were supported by an original theoretical analysis, demonstrated that the measurements of the tether force and angle between tethers at coalescence directly yield the bending rigidity, kappa, and the membrane tension, sigma, of the vesicles. Contrary to other methods used to probe the bending rigidity of vesicles, the proposed approach permits a direct measurement of kappa without requiring any control of the membrane tension. Finally, after validation of the method and proposal of possible applications, we experimentally investigated the dynamics of the coalescence process.