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We determine all Finsler metrics of Randers type for which the Riemannianpart is a scalar multiple of the Euclidean metric, on an open subset of theEuclidean plane, whose geodesics are circles. We show that the Riemannian partmust be of constant Gaussian curvature, and that for every such Riemannianmetric there is a class of Randers metrics satisfying the condition, determinedup to the addition of a total derivative, depending on a single parameter. Asone of several applications we exhibit a Finsler metric whose geodesics are theoriented horocycles in the Poincare disk, in each of the two possibleorientations.

A class of Finsler surfaces whose geodesics are circles