  We study percolation and the random cluster model on the triangular lattice
with 3-body interactions. Starting with percolation, we generalize the
star--triangle transformation: We introduce a new parameter (the 3-body term)
and identify configurations on the triangles solely by their connectivity. In
this new setup, necessary and sufficient conditions are found for positive
correlations and this is used to establish regions of percolation and
non-percolation. Next we apply this set of ideas to the $q>1$ random cluster
model: We derive duality relations for the suitable random cluster measures,
prove necessary and sufficient conditions for them to have positive
correlations, and finally prove some rigorous theorems concerning phase
transitions.
