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The Brachistochrone problem of mass-point moving in the vertical plane driven by gravity, viscous nonlinear drag with inequality constraint for the slope angle is considered. The normal component of the reaction force is a control variable. Principle maximum procedure allows to reduce the optimal control problem to the boundary value problem for a systems of two nonlinear differential equations. The extremal control is designed in a feedback form depending on the state variables and constraints. The optimal control structure is determined. It is shown that extremal trajectory includes no more than single arc with the motion along the constraint for the case of zero-drag and no more than two arcs with the motion along the constraint for the case of viscous drag.