From Thales Theorem to octic curves:

Locus computation is an essential issue in mathematics education, and a traditional feature of Dynamic Geometry software (DGS). The rising of programs merging DGS and Computer Algebra software (CAS) has fostered a combined approach to locus computation, quite performing in standard examples, but demanding an extended theoretical, and the related algorithmic counterpart, able to deal with less conventional situations. Here we formulate—and reflect about, yielding some proposals—on a few pending issues related to the protocols for the computation of parametric families of loci. Then we focus on a different source of difficulties, through the example of the very elementary and classical theorem of Thales. Thus, in the framework of the current development of automated deduction in geometry (ADG) tools, we will show how the automatic discovery of Thales's converse statement might require a locus computation that gives rise to an unexpected family of octic curves. Finally, we will exhibit how the handling (finding the equation, plotting, geometric characterization, etc.) of such curves requires the concourse of DGS and CAS programs, a mixed graphic–symbolic–numeric approach, and human–machine interaction, a cooperation that could be the basis towards achieving the required improvements concerning locus computation software.