Statistical geometric methods based on nearest-neighbor distributions are used, in connection with hard-particle systems, to develop Ornstein-Zernlike-like equations that have already been of considerable value in the statistical thermodynamic analysis of such systems and that promise to have even greater value. In this paper, we use these equations to (1) develop a relation that is valid for a hard particle system in unconstrained equilibrium and that shows that the insertion probability cannot vanish (short of closepacking) in such a system, (2) study the still incompletely settled issue concerning the equality of the hard-particle densities on the peripheries of cavities which are and are not occupied by hard particles and, in so doing, arrive at a relation that holds in a system in stable equilibrium but fails in a metastable system, (3) provide insight into the geometric mechanism of hard-particle phase transitions and allow simple estimates of the freezing densities, and (4) suggest a new physical interpretation for the direct correlation function.