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A corner is defined to be the vertex of a strict cone locally containing the support of a truss and having locally empty intersection with the support of the applied force. A topological perturbation called corner cutting is defined in two dimensions and the process is shown to have a negative effect on the mass of planar trusses, independent of the angle of the corner. Minimal, finite, planar trusses are therefore free of corners. Applied point forces are shown to be balanced by at least one truss, thereby showing that the admissible class is nonempty. Explicit mass and geometric bounds are presented as part of this construction.