  We study ground-state properties of a triangular triple quantum dot connected
to two superconducting (SC) leads. In this system orbital motion along the
triangular configuration causes various types of quantum phases, such as a
Kondo effect with a four-fold degenerate state and the Nagaoka ferromagnetic
mechanism, depending on the electron filling. The ground state also evolves as
the Cooper pairs penetrate from the SC leads. We describe the phase diagram in
a wide range of the parameter space, varying the gate voltage, the couplings
between the dots and leads, and also the Josephson phase between the SC gaps.
The results are obtained in the limit of large SC gap, carrying out exact
diagonalization of an effective Hamiltonian. We also discuss a classification
of the quantum states according to the fixed point of the Wilson numerical
renormalization group (NRG). Furthermore, we show that the Bogoliubov
zero-energy excitation determines the ground state of a $\pi$ Josephson
junction at small electron fillings.
