  We study the properties of the space of thermodynamic equilibrium states of
the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole in (2+1)-gravity. We use the
formalism of geometrothermodynamics to introduce in the space of equilibrium
states a $2-$dimensional thermodynamic metric whose curvature is non-vanishing,
indicating the presence of thermodynamic interaction, and free of
singularities, indicating the absence of phase transitions. Similar results are
obtained for generalizations of the BTZ black hole which include a Chern-Simons
term and a dilatonic field. Small logarithmic corrections of the entropy turn
out to be represented by small corrections of the thermodynamic curvature,
reinforcing the idea that thermodynamic curvature is a measure of thermodynamic
interaction.
