A numerical method for the reconstruction of interfaces in finite volume schemes for multiphase flows is presented. The computation of the triple point at the intersection of three materials in two dimensions of space is addressed. The determination of the normal vectors between pairs of materials is obtained with a finite element approximation. A numerical method for the localization of a triple point is described as the minimum of a constrained minimization problem inside an interfacial cell of the discretization. For given volume fractions of materials in the cell, an interior-point/Newton method is used for the reconstruction of the local geometry and the localization of the triple point. Initialization of the Newton method is performed with a derivative-free algorithm. Numerical results are presented for static and pure advection cases to illustrate the efficiency and robustness of the algorithm.