  Let M be ternary, homogeneous and simple. We prove that if M is finitely
constrained, then it is supersimple with finite SU-rank and dependence is
$k$-trivial for some $k < \omega$ and for finite sets of real elements. Now
suppose that, in addition, M is supersimple with SU-rank 1. If M is finitely
constrained then algebraic closure in M is trivial. We also find connections
between the nature of the constraints of M, the nature of the amalgamations
allowed by the age of M, and the nature of definable equivalence relations. A
key method of proof is to "extract" constraints (of M) from instances of
dividing and from definable equivalence relations. Finally, we give new
examples, including an uncountable family, of ternary homogeneous supersimple
structures of SU-rank 1.
