  We derive the BPS type of first order differential equations for the rotating
black hole solutions in the three-dimensional Einstein gravity coupled
minimally with a self-interacting scalar field, using fake supersymmetry
formalism. It turns out that the formalism is not complete and should be
augmented by an additional equation to imply the full equations of motion. We
identify this additional equation as a constraint by using an effective action
method. By computing the renormalized boundary stress tensor, we obtain the
mass and angular momentum of the black hole solutions of these first order
equations and confirm that they saturate the BPS bound.
