In this paper, we investigate weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. We show that the addition of weights to an arbitrary triangle mesh provides a geometric construction of an orthogonal dual structure obtained by displacing the primal mesh's circumcenters along a gradient vector field. From this flexible, yet principled definition of a dual we derive a discrete Laplace-Beltrami operator that preserves core properties of its continuous counterpart. We also propose several metric representations of these primal-dual structures for numerical convenience. In the process, we relate our work to the circle and sphere packing literature, and uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate that weighted triangulations offer a valuable extension to pairwise, intrinsic, and weighted Delaunay triangulations for the design of efficient and accurate computational tools useful in a variety of geometry processing tasks.