Figural concepts in proving by contradiction

Geometrical thinking in proving by contradiction involves specific and complex processes that can be source of difficulties for students. The goal of this article is to investigate on proof by contradiction in geometry, with particular emphasis on processes related to the treatment of the geometrical figures. The analysis, carried out with the lens of the Figural Concepts and Cognitive Unity frameworks, reveals that students, in order to conclude a proof by contradiction, need to restore the rupture between figural and conceptual components, and to try to give a geometrical meaning to the contradiction. Therefore, if in a proof by contradiction the involved geometrical figures have to be rejected (having deduced a contradiction), in the indirect argumentation proposed by many students, the figures are modified so that they do not appear absurd and impossible.