  We prove that there is a unique convex non-collinear central configuration of
the planar Newtonian four-body problem when two equal masses are located at
opposite vertices of a quadrilateral and, at most, only one of the remaining
masses is larger than the equal masses. Such central configuration posses a
symmetry line and it is a kite shaped quadrilateral.
  We also show that there is exactly one convex non-collinear central
configuration when the opposite masses are equal. Such central configuration
also posses a symmetry line and it is a rhombus.
