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This paper is concerned with applications of Tutte's barycentric embedding theorem (Proc. London Math. Soc. 13 (1963) 743-768). It presents a method for building isotopies of triangulations in the plane, based on Tutte's theorem and the computation of equilibrium stresses of graphs by Maxwell-Cremona's theorem; it also provides a counterexample showing that the analogue of Tutte's theorem in dimension 3 is false.