Computer-Generated Geometry Proofs in a Learning Context

Given its formal, logical, and spatial properties, geometry is well suited to teaching environments that include dynamic geometry systems (DGSs) , geometry automated theorem provers (GATPs), and repositories of geometric problems. These tools enable students to explore existing knowledge in addition to creating new constructions and testing new conjectures. In this chapter, we trace the evolution of current automatic proving technologies, how these technologies are beginning to be used by geometry practitioners in general to validate geometric conjectures and generate proofs with natural language and visual rendering, and foresee their evolution and applicability in an educational setting.

[1]  Pedro Quaresma,et al.  The full angle method on the OpenGeoProver , 2013, CICM Workshops.

[2]  Xiao-Shan Gao,et al.  Automated generation of readable proofs with geometric invariants , 2004, Journal of Automated Reasoning.

[3]  Xiao-Shan Gao,et al.  Automated Reasoning in Geometry , 2001, Handbook of Automated Reasoning.

[4]  Michael De Villiers.  Some pitfalls of dynamic geometry software , 2006 .

[5]  Thierry Coquand,et al.  Automating Coherent Logic , 2005, LPAR.

[6]  Xiao-Shan Gao,et al.  Visually Dynamic Presentation of Proofs in Plane Geometry , 2009, Journal of Automated Reasoning.

[7]  Sana Stojanovic,et al.  A Coherent Logic Based Geometry Theorem Prover Capable of Producing Formal and Readable Proofs , 2010, Automated Deduction in Geometry.

[8]  L. Hongbo,et al.  Clifford algebra approaches to mechanical geometry theorem proving , 2000 .

[9]  R. Nelsen.  Proofs Without Words: Exercises in Visual Thinking , 2020 .

[10]  Marc Bezem,et al.  A Vernacular for Coherent Logic , 2014, CICM.

[11]  Tomás Recio,et al.  An Introduction to Automated Discovery in Geometry through Symbolic Computation , 2012 .

[12]  Pedro Quaresma,et al.  System Description: GCLCprover + GeoThms , 2006, IJCAR.

[13]  Xiao-Shan Gao,et al.  An Introduction to Java Geometry Expert - (Extended Abstract) , 2008, Automated Deduction in Geometry.

[14]  Ke Wang,et al.  Automated Geometry Theorem Proving for Human-Readable Proofs , 2015, IJCAI.

[15]  Pedro Quaresma,et al.  Integrating Dynamic Geometry Software, Deduction Systems, and Theorem Repositories , 2006, MKM.

[16]  Xiao-Shan Gao,et al.  Visually Dynamic Presentation of Proofs in Plane Geometry , 2010, Journal of Automated Reasoning.

[17]  Tomás Recio,et al.  Automated Theorem Proving in GeoGebra: Current Achievements , 2015, Journal of Automated Reasoning.

[18]  Pedro Quaresma,et al.  The Web Geometry Laboratory Project , 2013, MKM/Calculemus/DML.

[19]  Pedro Quaresma,et al.  Web geometry laboratory: case studies in Portugal and Serbia , 2018, Interact. Learn. Environ..

[20]  Pedro Quaresma,et al.  Automatic Verification of Regular Constructions in Dynamic Geometry Systems , 2006, Automated Deduction in Geometry.

[21]  Deepak Kapur,et al.  Using Gröbner Bases to Reason About Geometry Problems , 1986, J. Symb. Comput..

[22]  Philippe R. Richard,et al.  The concept of proof in the light of mathematical work , 2016 .

[23]  G. Hanna,et al.  Visualisation and proof: a brief survey of philosophical perspectives , 2007 .

[24]  Sara Hennessy,et al.  Constructions of dynamic geometry: A study of the interpretative flexibility of educational software in classroom practice , 2008, Comput. Educ..

[25]  Predrag Janicic,et al.  GCLC - A Tool for Constructive Euclidean Geometry and More Than That , 2006, ICMS.

[26]  Pedro Quaresma,et al.  WGL, a web laboratory for geometry , 2017, Education and Information Technologies.

[27]  Dongming Wang,et al.  Reasoning about Geometric Problems using an Elimination Method , 1995 .

[28]  Jianguo Jiang,et al.  A review and prospect of readable machine proofs for geometry theorems , 2012, J. Syst. Sci. Complex..

[29]  Xiao-Shan Gao,et al.  Automated generation of readable proofs with geometric invariants , 1996, Journal of Automated Reasoning.

[30]  Pedro Quaresma,et al.  Towards an Intelligent and Dynamic Geometry Book , 2017, Math. Comput. Sci..

[31]  Pedro Quaresma,et al.  The Area Method , 2010, Journal of Automated Reasoning.

[32]  Jürgen Richter-Gebert,et al.  The Interactive Geometry Software Cinderella , 2000 .

[33]  Gila Hanna,et al.  Proof, Explanation and Exploration: An Overview , 2000 .

[34]  Tomás Recio,et al.  Towards the Automatic Discovery of Theorems in GeoGebra , 2016, ICMS.

[35]  Zoltán Kovács.  The Relation Tool in GeoGebra 5 , 2014, ADG.

[36]  Pedro Quaresma,et al.  Integrating DGSs and GATPs in an Adaptative and Collaborative Blended-Learning Web-Environment , 2011, THedu.