Following our earlier works [1,2], we introduce a new simple metric form for
thermodynamic geometry. The new thermodynamic geometry (NTG) indicates
correctly a one-to-one correspondence between curvature singularities and phase
transitions. For a non-homogeneous thermodynamic potential, by considering a
phantom RN-AdS black hole, the NTG formalism represents a one-to-one
correspondence between curvature singularities and phase transitions. Working
with the NTG metric neatly excludes all unphysical points that were generated
in other geometric formulations of thermodynamics such as the
geometrothermodynamics (GTD). We show that the NTG is conformally related to
GTD. However, the conformal transformation is singular at unphysical points
that were generated by the GTD.