We propose an original method to synthesize the dimensions of a planar mechanism (linkage) whose function is to generate a trajectory shape. Most graphical and analytical synthesis methods for path generators require specifying the desired trajectory in a non-functional manner, by a list of points rather than a pure shape. Concerning the dimensional optimization methods, they turn out to be slow and their convergence depends on the initial solution. Alternatively, we propose a case-based approach (i.e., couples of trajectories and dimensions of a given structure mechanism) using a neural network. The first stage consists in the generation of a huge case number through kinematic simulations, for random values of dimensions, and in a learning process of the neural network. In the second stage, of utilization, the neural network instantaneously makes it possible to obtain an approximate solution of the synthesis problem, which is an interpolation of close cases. We show on the four-bar linkage example the good quality of the synthesized solutions, for a tiny size of the network. Next, these solutions may be used as judicious initial solutions for a conventional dimensional optimization.