  The triangle distribution function f^(3) for three mutual nearest neighbors
in the plane describes basic aspects of short-range order and statistical
thermodynamics in two-dimensional many-particle systems. This paper examines
prospects for constructing a self-consistent calculation for the rigid-disk
system f^(3). We present several identities obeyed by f^(3). A rudimentary
closure suggested by scaled-particle theory is introduced. In conjunction with
three of the basic identities, this closure leads to a unique f^(3) over the
entire density range. The pressure equation of state exhibits qualitatively
correct behavior in both the low density and the close-packed limits, but no
intervening phase transition appears. We discuss extensions to improved disk
closures, and to the three-dimensional rigid sphere system.
