The themes introduced in this chapter illustrate some important aspects of geometrical construction critical to correctly posing the problem of the relationships between geometry and cognition (cf. the following chapter). In the history of philosophy there are at least three main ways for designing the role of geometrical construction in hypothesis generation, always considered in the perspective of problem solving performances. All aim at demonstrating that the activity of geometrical construction is paradoxical, either illusory or obscure, implicit, and not analyzable.