  It is shown that quadratic constraints are compatible with the geometric
integrability scheme of the multidimensional quadrilateral lattice equation.
The corresponding Ribaucour reduction of the fundamental transformation of
quadrilateral lattices is found as well, and superposition of the Ribaucour
transformations is presented in the vectorial framework. Finally, the quadratic
reduction approach is illustrated on the example of multidimensional circular
lattices.
