  We consider bound states of asymmetric three-body systems confined to two
dimensions. In the universal regime, two energy ratios and two mass ratios
provide complete knowledge of the three-body energy measured in units of one
two-body energy. The lowest number of stable bound states is produced when one
mass is larger than two similar masses. We focus on selected asymmetric systems
of interest in cold atom physics. The scaled three-body energy and the two
scaled two-body energies are related through an equation for a supercircle
whose radius increases almost linearly with three-body energy. The exponents
exhibit an increasing behavior with three-body energy. The mass dependence is
highly non-trivial. We give a simple relation that predicts the universal
three-body energy.
