New tools in GeoGebra offering novel opportunities to teach loci and envelopes

Abstract: GeoGebra is an open source mathematics education software tool being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians and programmers helping the GeoGebra developer team. The joint work, based on former researches, started in 2010 and continued until present days, now enables fast locus and envelope computations even in a web browser in full HTML5 mode. Thus, classroom demonstrations and deeper investigations of dynamic analytical geometry are ready to use on tablets or smartphones as well. In our paper we consider some typical secondary school topics where investigating loci is a natural way of defining mathematical objects. We discuss the technical possibilities in GeoGebra by using the new commands LocusEquation and Envelope, showing through different examples how these commands can enrich the learning of mathematics. The covered school topics include definition of a parabola and other conics in different situations like synthetic definitions or points and curves associated with a triangle. Despite the fact that in most secondary schools, no other than quadratic curves are discussed, simple generalization of some exercises, and also every day problems, will smoothly introduce higher order algebraic curves. Thus our paper mentions the cubic curve "strophoid" as locus of the orthocenter of a triangle when one of the vertices moves on a circle. Also quartic "cardioid" and sextic "nephroid" can be of every day interest when investigating mathematics in, say, a coffee cup. We also focus on GeoGebra specific tips and tricks when constructing a geometric figure to be available for getting the locus equation. Among others, simplification and synthetization (via the intercept theorem) are mentioned.