This work characterizes and classifies the geometric configurations that can arise in the relative equilibria of four point vortices on the unbounded plane. The problem is approached through a two-dimensional parametric formulation, enabling the identification of fourth vortex locations that yield relative equilibria for a given triangular arrangement of the first three vortices. Building on the configuration matrix approach, a bivariate polynomial is derived from the matrix minors such that its roots determine the admissible positions of the fourth vortex. In addition, a detailed analytical description of how the root structure of this polynomial varies with the two parameters defining the vortex triangle is also presented. The results show that the fourth vortex locations generally form a one-dimensional continuum, consisting of a disconnected pair of bounded and unbounded curves. Several important families of relative equilibria are identified by tracing how the two curves evolve across the parameter space. The analytical results, supported by numerical examples, offer a comprehensive geometric description of the configuration space and its bifurcation structures.

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• Vortex dynamics
• Vortex interactions
• Vortex lattices
• Hamiltonian systems