We improve the complex number identity proving method to a fully automated procedure, based on elimination ideals. By using declarative equations or rewriting each real-relational hypothesis $h_i$ to $h_i-r_i$, and the thesis $t$ to $t-r$, clearing the denominators and introducing an extra expression with a slack variable, we eliminate all free and relational point variables. From the obtained ideal $I$ in $\mathbb{Q}[r,r_1,r_2,\ldots]$ we can find a conclusive result. It plays an important role that if $r_1,r_2,\ldots$ are real, $r$ must also be real if there is a linear polynomial $p(r)\in I$, unless division by zero occurs when expressing $r$. Our results are presented in Mathematica, Maple and in a new version of the Giac computer algebra system. Finally, we present a prototype of the automated procedure in an experimental version of the dynamic geometry software GeoGebra.

In this article, we solve some of the geometry problems of the N\'aboj 2023 competition with the help of a computer, using examples that the software tool GeoGebra Discovery can calculate. In each case, the calculation requires symbolic computations. We analyze the difficulty of feeding the problem into the machine and set further goals to make the problems of this type of contests even more tractable in the future.

Automated reasoning in geometry is available in various software tools for several years, mostly in prover packages. In this paper we pay our attention to a non-trivial presence of a geometry prover in the software tool GeoGebra Discovery [4, 5] that aims at reaching secondary schools with its intuitive user interface. Most importantly, we give a report on a STEM/STEAM project that was discussed in a group of prospective mathematics teachers at the Private University College of Education of the Diocese of Linz in Upper Austria during the winter semester 2022/23, in the frame of a course that focuses on exploiting technology in mathematics education (36 students in 2 working groups). This project consisted of several other experiments that were already communicated by the second author. The discussed activity, a detailed study of the movement of a rocking camel, is however, completely new. Also, some major improvements in the underlying software tool (implemented by the second author with a substantial help of the students' feedback), makes it much easier to model similar project setups and conclude mathematical knowledge in an automated way.

We address, through the automated reasoning tools in GeoGebra Discovery, a problem from a regional phase of the Austrian Mathematics Olympiad 2023. Trying to solve this problem gives rise to four different kind of feedback: the almost instantaneous, automated solution of the proposed problem; the measure of its complexity, according to some recent proposals; the automated discovery of a generalization of the given assertion, showing that the same statement is true over more general polygons than those mentioned in the problem; and the difficulties associated to the analysis of the surprising and involved high number of degenerate cases that appear when using the LocusEquation command in this problem. In our communication we will describe and reflect on these diverse issues, enhancing its exemplar role for showing some of the advantages, problems, and current fields of development of GeoGebra Discovery.

See [1] for a general description and references. The goal of the current contribution is to present some ongoing work regarding two different, but related, important improvements of GeoGebra Discovery. One, to visualize the different steps that GG Discovery performs with a given geometric statement until it declares its truth (or failure). Two, to test, through different elementary examples, the suitability of an original proposal to evaluate the interest, complexity or difficulty of a given statement. Let us advance that our proposal involves the notion of syzygy of a set of polynomials. The relevance of showing details about each of the steps performed by our automated reasoning algorithms implemented in GG Discovery is quite evident. In fact, as a consequence of the result in [2], describing the formalization of the arithmetization of Euclidean plane geometry, proofs of geometric statements obtained using algebraic geometry algorithms are also valid on the synthetic geometry realm.

We describe a prototype of a new experimental GeoGebra command and tool Discover that analyzes geometric figures for salient patterns, properties, and theorems. This tool is a basic implementation of automated discovery in elementary planar geometry. The paper focuses on the mathematical background of the implementation, as well as methods to avoid combinatorial explosion when storing the interesting properties of a geometric figure.