The Quadratic Assignment Problem is computationally one of the most dilficult NP-Hard problems. Recently, in 1996, Oguz introduced the so called ’triangle constraints’ into an extended model of the Travelling .Salesman problem (TSP) in an attempt to give a full description of the TSP polytope. In this study, we make use of these constraints in the context of the (Quadratic Assignment Problem (Q.AP). We discuss the relationships between the polytopes defined by the formulations with and without triangle constraints and we provide necessary and sufficient conditions for these constraints to give a full description of the QAP polytope. by-product of our analysis is that the triangle constraints suffice to define the QAP poly tope for n = 4.