  In this paper, we study the properties and thermodynamic stability of the
plane symmetry black hole from the viewpoint of geometry. Weinhold metric and
Ruppeiner metric are obtained, respectively. The Weinhold curvature gives phase
transition points, which correspond to the first-order phase transition only at
N=1, where $N$ is a parameter in the plane symmetry black hole. While the
Ruppeiner one shows first-order phase transition points for arbitrary $N\neq
1$. Both of which give no any information about the second-order phase
transition. Considering the Legendre invariant proposed by Quevedo et. al., we
obtain a unified geometry metric, which gives a correctly the behavior of the
thermodynamic interactions and phase transitions. The geometry is also found to
be curved and the scalar curvature goes to negative infinity at the Davies'
phase transition points when the logarithmic correction is included.
