  In the present paper, we consider two applications of the pentagon equation.
The first deals with actions of flips on edges of triangulations labelled by
rational functions in some variables. The second can be formulated as a system
of linear equations with variables corresponding to triangles of a
triangulation. The general method says that if there is some general {\em data}
(say, edge lengths or areas) associated with {\em states} (say, triangulations)
and a general {\em data transformation rule} (say, how lengths or areas are
changed under flips) then after returning to the initial state we recover the
initial data.
