  We work towards the general solution of the two-body problem in
2+1-dimensional general relativity with a negative cosmological constant. The
BTZ solutions corresponding to black holes, point particles and overspinning
particles can be considered either as objects in their own right, or as the
exterior solution of compact objects with a given mass $M$ and spin $J$, such
as rotating fluid stars. We compare and contrast the metric approach to the
group-theoretical one of characterising the BTZ solutions as identifications of
2+1-dimensional anti-de Sitter spacetime under an isometry. We then move on to
the two-body problem. In this paper, we restrict the two objects to the point
particle range $|J|-1\le M<-|J|$, or their massless equivalents, obtained by an
infinite boost. (Both anti-de Sitter space and massless particles have $M=-1$,
$J=0$). We derive analytic expressions for the total mass $M_\text{tot}$ and
spin $J_\text{tot}$ of the system in terms of the six gauge-invariant
parameters of the two-particle system: the rest mass and spin of each object,
and the impact parameter and energy of the orbit. Based on work of Holst and
Matschull on the case of two massless, nonspinning particles, we conjecture
that the black hole formation threshold is $M_\text{tot}=|J_\text{tot}|$. The
threshold solutions are then extremal black holes. We determine when the global
geometry is a black hole, an eternal binary system, or a closed universe.
