  We exploit the idea that the character of an irreducible finite dimensional
$\mathfrak{gl}_n$-module is the sum of certain exponents of integer points in a
Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In
order to show how the result of such a calculation matches Weyl's character
formula we prove some interesting combinatorial traits of Gelfand-Tsetlin
polytopes. Namely, we show that under the relevant substitution the integer
point transforms of all but $n!$ vertices vanish, the remaining ones being the
summands in Weyl's formula.
