Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the
q-state quantum Potts model on the square lattice with $q\ensuremath{\in}{2,3,4,5}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2
XXX and anisotropic
XYX models in an external magnetic field, and the antiferromagnetic spin-1
XXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.