We discuss the origin of the Wilson polygon-MHV amplitude duality at a perturbative level. It is shown that the duality for the MHV amplitudes at the one-loop level can be proven upon a particular change of variables in Feynman parametrization and with the use of the relation between Feynman integrals at different space-time dimensions. Some generalization of the duality which implies the insertion of a particular vertex operator at the Wilson triangle is found for the 3-point function. We discuss the analytical structure of Wilson loop diagrams and present the corresponding Landau equations. The geometrical interpretation of the loop diagram in terms of the hyperbolic geometry is discussed.