Multiphase contact lines are present in a number of practical cases, and they are also interesting from a theoretical perspective. The equilibrium relation for a multiphase contact line and multiphase contact point is presented using a thermodynamic free-energy formulation. Calculus of variation is applied to the free-energy functional of a multiphase system, and vectorial mechanical equilibrium conditions for contact lines and points are derived. The developed relation for the multiphase contact line could be regarded as the generalization of the classical Neumann triangle relation. It is shown that the mechanical equilibrium condition for a multiphase contact line may graphically be represented by a planar polygon, whose sides consist of the surface and line tensions. The stated mechanical equilibrium condition for a multiphase contact point can be regarded as the zeroth-order analogue of the Laplace equation (second order) and Neumann triangle relation (first order). It is also shown that the mechanical equilibrium condition for a multiphase contact point may graphically be represented by a nonplanar polygon, whose sides consist of the line tensions. To demonstrate the application of the above relations, a model lamina system is discussed in terms of the stability of the multiphase contact lines.