Focusing attention on auxiliary lines when introduced into geometric problems

The introduction of auxiliary elements as a method of solving problems in high-school geometry is considered here from two perspectives: first, to elicit recalling some known result or concretizing a definition and, second, as a means of shifting the focus and structure of the students’ attention. We present and compare various examples of how auxiliary elements can be introduced in various problems and proofs and characterize their auxiliary quality. Some auxiliary elements unite previously unrelated components of the original diagram; others divide a given complex entity into manageable ones. Implications for further educational research and mathematics instruction are proposed.