25 January 2021

With today's release, GeoGebra Discovery supports comparison of algebraic expressions in an improved way. To illustrate this new feature let us imagine a planar construction which is invariant of scaling, that is, by dragging its free points similar figures will be obtained for every position of the free points. In fact, if only compass-and-straightedge steps are allowed, this is usually the case.

For example, given a square. We want to compare the length of its diagonals (in fact they are equal long) to the length of the side of the square. It is clear that for all positions of the two free input points (which designate one side of the square) the ratio of a diagonal and a side is \(\sqrt2\).

It is very simple to try to get this result symbolically: Just select the Relation tool and select segments \(f\) and \(j\), then, by clicking on More..., you get the required proportion. The same result can be obtained by typing the command Relation(f,j). In some cases, however, we are interested in more complicated relations. Here, for example, a student may ask the relationship between, say, \(f\cdot g\) and \(j\).

GeoGebra requires some tricks here in its internals. First of all, it is clear that \(f\cdot g\) is a quadratic quantity, because it is a product of two linear quantities. On the other hand, \(j\) is still a linear quantity. So any correspondence between the quadratic and linear quantities must be related to some change of the linear expression, namely, its square should be observed. When asking Relation(f·g,j), GeoGebra will compare \(f\cdot g\) and \(j^2\), by symbolically searching for a matching constant \(K\in\mathbb{R}^+\) such that \[f\cdot g=K\cdot j^2.\]

Let us mention some possible uses. A simple way to do some more experiments in the square above is to ask Relation(f^2+g^2,j), or, being a bit more unusual, Relation(f^3+g^3,j^2). Or even Relation(f^3+j^3,g^2). (Try it! You can use copy-paste to enter this formula in GeoGebra.)

Some other experiments can be done in the following way: In a regular pentagon the product of the lengths of the diagonals from a given vertex can be expressed with the formula \(f\cdot k\cdot l\cdot j\), see below:

At the same time we can create the circumcircle of the pentagon and consider its radius \(R\). Now: what is the relationship between the above product and \(R\)? Can we generalize this result?

Acknowledgments. As usually, this is joint work with some of my colleagues. Now much credits go to Róbert Vajda, Tomás Recio, M. Pilar Vélez and Lajos Szilassi.