The contact line quadrilateral relation is a generalization of the classical Neumann triangle relation that includes the effect of line tension. The relation is derived using a minimum free energy principle that may be graphically represented by a quadrilateral whose four sides consist of three surface tension quantities and a line tension term. When the line tension term vanishes, the quadrilateral relation simplifies to the classical Neumann triangle relation. Applications of the quadrilateral relation are illustrated using several axisymmetric capillary geometries.