  We study collinear relative equilibria of the planar four-vortex problem
where three of the four vortex strengths are identical. The $S_3$ invariance
obtained from the equality of vorticities is used to reduce the defining
equations and organize the solutions into two distinct groups based on the
ordering of the vortices along the line. The number and type of solutions are
given, along with a discussion of the bifurcations that occur. The linear
stability of all solutions is investigated rigorously and stable solutions are
found to exist for cases where the vorticities have mixed signs. We employ a
combination of analysis and computational algebraic geometry to prove our
results.
