Using Dynamic Geometry Software to Gain Insight into a Proof

Abstract

The ability of a student to obtain, analyze, measure and compare the many instances of a mathematical proposition in a dynamic geometry environment gives him/her opportunities to make conjectures and test a proposition. These roles for dynamic geometry software are widely acknowledged as having the potential to enrich the teaching of geometry. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by ‘‘what if’’ and ‘‘what if not’’ questions, has the potential to promote links between empirical and deductive reasoning. A classic set of problems in Euclidean plane geometry consists of finding the path of a point that is subject to given constraints. Its systematic study goes back to a lost work (in two books) of Apollonius of Perga, Plane loci (Botana and Valcarce 2003). Except for the simplest loci, such as lines, circles and perhaps the conics, this subject is usually avoided in most geometry texts. This is due to the common difficulties faced when mentally visualizing various objects with different movements. With the emergence of dynamic geometry software, the locus problems have attracted new interest from researchers. Using the locus generation features of these software one can easily obtain the locus of a point under some constraints. Although there are some minor differences, different dynamic geometry software, behave in a similar way when obtaining loci. In general, two objects must be selected: the first one, usually a point called the driving point or mover, is bound to a path, whereas the other, the locus point, must depend somehow on the first one. Since the element