We propose a reformulation of the Faltings-Wüstholz
nonlinear version of Schmidt's subspace theorem with the help of toric
deformations and Chow polytopes. Moreover, we show that the arithmetic
Bézout theorem in Arakelov geometry can be used to obtain a
Bézout theorem for Mumford's degree of contact. This is a
birational invariant often considered in geometric invariant theory
(GIT). The originality of this last result relies on the
interpretation of GIT as a degeneration of Arakelov geometry. This
should enable us to transfer all known results of Arakelov geometry
into GIT.