Lipid monolayer domains at the air-water interface can have a wide variety of shapes, determined by the energy balance between line tension and long-range dipole interactions. The present work focuses on the transition from a circle-the equilibrium shape for strong line tension-to a distorted, dogbonelike shape with 2-fold symmetry. This transition takes place when the line tension drops below a critical value. From numerical calculations, it is found that the transition is weakly first-order. To simplify the numerical work, a novel method is introduced to deal with the cutoff parameter DELTA. It is shown how DELTA can be absorbed in a redefinition of the line tension. Finally, a simple scaling law for the energy of lipid domains as a function of domain size is derived.