We consider three microscopic model molecular systems, each containing an equimolar mixture of a chiral molecule and its nonsuperimposable mirror image. The molecules in each model are assumed to lie on a thin film in such a way that they occupy the sites of a honeycomb lattice. Although neither enantiomorph is externally favored at low temperatures, we prove that for one range of interactions, chiral segregation into ordered phases containing a single enantiomorph occurs for two of the models and, in a second range of interactions, ordered racemic phases (containing equal numbers of each enantiomorph) occur for the two models. For a third range of interactions, each of the two models has an infinite number of ground-state configurations and, moreover, an associated residual entropy. In all three ranges of interactions considered, the third model has an infinite number of ground-state configurations and a residual entropy.