Planar scenes predominate in man-made environments, e.g.interior or facades of buildings and in ground images from aerial vehicles. Points lying on those surfaces can be reconstructed from their observations in two images. However, generic reconstruction algorithms output 3D points not lying on the plane, thus obtaining inaccurate reconstructions. The problem also turns to be non-convex with many local minima, hence hindering the performance of iterative method. Therefore, being able to obtain and certify the optimal solution to this problem is of special relevant for these applications. In this paper we first propose a fast and certifiable algorithm that both estimates and certifies the optimal solution to the triangulation problem. From this formulation, we also present an optimality certificate that tells us whether a given solution (obtained by any solver) is the global optimum. Last, from this certificate we derive a sufficient (but not necessary) optimality condition that allows us to certify optimality in less than one microsecond. We test the proposed algorithms on extensive experiments on both synthetic and real data. Code is made available at https://github.com/mergarsal.