<p>A new methodological approach to the statement and solution of the variational problem of the brachistochrone on an inclined plane is proposed. This approach takes into account the joint minimization of the time of motion of a material point on the inclined plane without friction and initial velocity in a vertical uniform gravitational field, and the length of the curve along which its brachistochrone motion is carried out. Therefore, a multiplicative criterion is created which allows for optimizing the product of two separate criteria for minimizing the time and length of the curve of brachistochrone motion of the material point. In the process of solving, a corresponding isoperimetric problem is stated and solved which allows for an analytical solution in closed form as the systems of parametric equations that algebraically describe the brachistochrone curves on the inclined plane. All possible types and variants of curves that arise in this problem depending on the boundary conditions and parameters of the inclined plane are considered. Based on the solution of the auxiliary problem and using a new multiplicative criterion, an optimal solution to the brachistochrone problem on the inclined plane is obtained.</p>