A continuing debate in elementary geometry: the Simson–Wallace line and its many generalisations

For some years now the importance has been appraised of demonstrating elementary geometry to pupils and future teachers through interactive geometry software. This fits within a view of the teaching of geometry that stresses a hands-on approach, thanks to which it is possible to teach the subject via historical syllabi, touching on ideas from different origins and of a transversal nature. The debate about the role of elementary geometry in the last 30 years is connected to this, with contributions by scholars such as Yaglom, Scimemi and Betti. In the perspective of following a sequence of elementary geometry constructions historically connected with each other, we suggest a path that analyses subjects linked with the Simson–Wallace line (and some significant points, such as the Clifford point); its history is full of intriguing ideas which in the past aroused the interest of great mathematicians as Steiner, Cremona and Clifford.