We consider the question of finding a periodic solution for the planar Newtonian N-body problem with equal masses, where each body is travelling along the same closed path. We provide a computer-assisted proof for the following facts: the local uniqueness and the convexity of the Chenciner and Montgomery Eight, the existence (and the local uniqueness) for Gerver's Super-Eight for 4-bodies and a doubly symmetric linear chain for 6-bodies.