  We investigate the symmetry of point vortices with one dominant vortex and
four vortices with infinitesimal circulations in the (1+4)-vortex problem, a
subcase of the five-vortex problem. The four infinitesimal vortices inscribe
quadrilaterals in the unit circle with the dominant vortex at the origin. We
consider symmetric configurations which have one degree of spacial freedom,
namely the (1+N)-gon, kites, rectangles, and trapezoids with three equal sides.
We show there is only one possible rectangular configuration (up to rotation
and ordering of the vortices) and one possible trapezoid with three equal sides
(up to rotation and ordering), while there are parametrically defined families
of kites. Additionally we consider the (1+4)-gon and show that the
infinitesimal vortices must have equal circulations on opposite corners of the
square. The proofs are heavily dependent on techniques from algebraic geometry
and require the use of a computer to calculate Grobner bases.
