In this article we study multiple steady states in ternary homogeneous azeotropic distillation. We show that in the case of infinite reflux and an infinite number of trays, multiple steady states exist when the distillate flow varies non-monotonically along the continuation path of the bifurcation diagram with the distillate flow as the bifurcation parameter. We derive a necessary and sufficient condition for the existence of these multiple steady states based on the geometry of the distillation region boundaries. We also locate in the composition triangle the feed compositions that lead to these multiple steady states. We further note that most of these results are independent of the thermodynamic model used. We show that the prediction of the existence of multiple steady states in the case of infinite reflux and an infinite number of trays has relevant implications for columns operating at finite reflux and with a finite number of trays. Using numerically constructed bifurcation diagrams for specific examples, we show that these multiplicities tend to vanish for small columns and/or for low reflux flows. Finally, we comment on the effect of multiplicities on column design and operation for some specific examples.