Equation of state of the rigid disk fluid from its triangle distribution

The distribution function $f(3)$ for triplets of mutual nearest neighbors offers a description of local order for many-particle systems confined to a plane. This paper proposes a self-consistent theory for $f(3)$ in the case of the classical rigid disk model, using three basic identities for closure. Numerical analysis of the resulting coupled nonlinear integral equations yields predictions for the pressure, the boundary tension, and the Kirkwood superposition defect for three disks in mutual contact. The approximation employed implicitly constrains the disk system to remain in the fluid phase at all densities up to close packing $(ρa2=2/31/2).$ The pressure and boundary tension agree reasonably well with the corresponding predictions of the two-dimensional scaled particle theory, but the former agrees even better with a rational approximant due to Sanchez that reproduces eight virial coefficients.