  Biasing a three-terminal Josephson junction (TTJ) with symmetrical voltages
$0,V,-V$ leads to new kinds of DC currents, namely quartet Josephson currents
and phase-dependent multiple Andreev reflection (MAR) currents. We study these
currents in a system where a normal diffusive metallic node $N$ is connected to
three terminals $S_{0,1,2}$ by barriers of arbitrary transparency. We use the
quantum circuit theory to calculate the current in each terminal, including
decoherence. In addition to the stationary combination
$\varphi_Q=\varphi_1+\varphi_2-2\varphi_0$ of the terminal phases $\varphi_i$,
the bias voltage $V$ appears as a new and unusual control variable for a DC
Josephson current. A general feature is the sign changes of the current-phase
characteristics, manifesting in minima of the quartet ``critical current".
Those sign changes can be triggered by the voltage, by the junction
transparency or by decoherence. We study the possible separation of quartet
currents from MAR currents in different regimes of parameters, including an
"funnel" regime with very asymmetric couplings to $S_{0,1,2}$. In the regime of
low transparency and asymetric couplings, we provide an analytic perturbative
expression for the currents which shows an excellent agreement with the full
numerical results.
