  We determine all Finsler metrics of Randers type for which the Riemannian
part is a scalar multiple of the Euclidean metric, on an open subset of the
Euclidean plane, whose geodesics are circles. We show that the Riemannian part
must be of constant Gaussian curvature, and that for every such Riemannian
metric there is a class of Randers metrics satisfying the condition, determined
up to the addition of a total derivative, depending on a single parameter. As
one of several applications we exhibit a Finsler metric whose geodesics are the
oriented horocycles in the Poincare disk, in each of the two possible
orientations.
