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In this paper we derive new two-dimensional (2-D) quadrature formulas for the discretization of boundary integral equations in the presence of conducting or dielectric edges. The proposed formulas allow us to exactly integrate polynomials of degree less than or equal to five, multiplied by an algebraic singular factor that diverges along one side of the triangular integration domain. This is the kind of singularity that occurs when physical edges are present in both conducting and dielectric bodies. Numerical tests are performed on the presented formulas, in order to validate the achieved improvement in accuracy, and examples are given of their application to the determination of radar cross-section of 3-D metallic objects.