Sliding with Friction and The Brachistochrone Problem

New formulas for a family of extreme curves are obtained that provide a minimum descent time between given points in a gravity field taking into account the force of dry sliding friction. Parametric equations for these curves depend on the friction coefficient and generalize the equation of the cycloid (brachistochrone for a smooth surface) to the case of sliding with friction. The results of numerical calculations in the program "Mathcad" prove that the found curves really provide a minimum descent time for the given parameters of the problem and can claim to be brachistochrone with friction.