  We establish a theory of optimal efficiency and power for three-terminal
thermoelectric engines which have two independent output electric currents and
one input heat current. This set-up goes beyond the conventional heat engines
with only one output electric current. For such a set-up, we derive the optimal
efficiency and power and their trade-off for three-terminal heat engines with
and without time-reversal symmetry. The formalism goes beyond the known optimal
efficiency and power for systems with or without time-reversal symmetry,
showing interesting features that have not been revealed before. A concrete
example of quantum-dot heat engine is studied to show that the current set-up
can have much improved efficiency and power compared with previous set-ups with
only one output electric current. Our analytical results also apply for
thermoelectric heat engines with multiple output electric currents, providing
an alternative scheme toward future high-performance thermoelectric materials.
