Students’ reasons for introducing auxiliary lines in proving situations

This paper focuses on reasons students’ give while introducing auxiliary lines in geometry proofs. Three cases of pairs of high school students participating in proving activities are presented. The overarching theme of the tasks proposed to the students is the comparison of areas of triangles and/or parallelograms. The students gave a wide spectrum of reasons when introducing an auxiliary line. Two main groups of reasons are discerned: The students introduced auxiliary lines recalling some known results or definitions and modified the given diagrams accordingly, as a part of a learned procedure. The students introduced auxiliary lines anticipating to receive more information from a modified situation. Students may combine reasons of recalling and anticipating nature when introducing an auxiliary line.