Some hidden harmonies between new and old geometric loci

The history of Mathematics is full of discoveries of geometric loci. Many of them began famous since they allowed the solution of important questions as, for example, those arising from the three famous problems of Greek geometry which were unsolvabled by Euclidean methods. So, the spiral of Archimedes, the concoid of Nicomedes, the cissoid of Diocles, and the quadratrix of Hippias were accorded full geometric status. For centuries the study of geometric loci occupied many of mathematical researches and by the XVII century the analitic geometry and the calculus allowed the discovery of a moltitude of curves. But the Mathematics is also a discipline looking to the past with new perspectives, so it is not unlikely that a mathematician can encounter some old geometric loci hidden into problems different from the primal ones. This new point of view can lay the foundations for new geometric discoveries and all that enlightens the mathematical landscape.