We present a method to assess the ranges of feasible separations in the distillation of ternary azeotrope-forming mixtures. The absolute boundaries for distillation in single- and multi-feed columns can be derived from a pinch curve analysis that locates the compositions of possible stationary points in each column section. The bifurcations of pinch trajectories explain the complex phenomena observable in homogeneous azeotropic distillation columns, in particular the crossing of boundaries for total reflux and the existence of a maximum reflux for such separations in single- and in multi-feed columns. These insights also lead to a simple new strategy to identify promising entrainers.