  We give a complete proof of Thurston's celebrated hyperbolic Dehn filling
theorem, following the ideal triangulation approach of Thurston and
Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always
exists, using only a partially flat one, obtained by subdividing an
Epstein-Penner decomposition. This forces us to deal with negatively oriented
tetrahedra. Our analysis of the set of hyperbolic Dehn filling coefficients is
elementary and self-contained. In particular, it does not assume smoothness of
the complete point in the variety of deformations.
