  We explore the properties of the equilibrium space of van der Waals
thermodynamic systems. We use an invariant representation of the fundamental
equation by using the law of corresponding states, which allows us to perform a
general analysis for all possible van der Waals systems. The investigation of
the equilibrium space is performed by using the Legendre invariant formalism of
geometrothermodynamics, which guarantees the independence of the results from
the choice of thermodynamic potential. We find all the curvature singularities
of the equilibrium space that correspond to first and second order phase
transitions. We compare our results with those obtained by using Hessian
metrics for the equilibrium space. We conclude that the formalism of
geometrothermodynamics allows us to determine the complete phase transition
structure of systems with two thermodynamic degrees of freedom.
