J Autom Reasoning (2009) 43:203–236 DOI 10.1007/s10817-009-9133-x On Protocols for the Automated Discovery of Theorems in Elementary Geometry G. Dalzotto · T. Recio Received: 4 June 2009 / Accepted: 7 June 2009 / Published online: 24 June 2009 © Springer Science + Business Media B.V. 2009 Abstract In this paper we consider the problem of dealing automatically with arbitrary geometric statements (including, in particular, those that are generally false) aiming to find complementary hypotheses for the statements to become true. Our approach proceeds within the framework of computational algebraic geometry. First we argue and propose a plausible protocol for automatic discovery, and then we present some algorithmic criteria, as well as the meaning (regarding the algebraic geometry of the varieties involved in the given statement), for the protocol success/failure. A detailed collection of examples in also included. Keywords Automatic theorem proving · Automatic theorem discovery · Elementary geometry · Computational algebraic geometry 1 Introduction In this paper we will deal with automatic discovery of elementary geometry theorems. We address this issue within the algebraic geometry framework that has already shown its success for automatic theorem proving. Roughly speaking, the algebraic geometry scheme towards theorem proving, proceeds translating theses and hypotheses about geometric entities into systems of polynomial equations, say T ={t 1 = 0,..., t s = 0} and H ={h 1 = 0,..., h r = 0}. Last author supported by grant “Algoritmos en Geometría Algebraica de Curvas y Superficies” (MTM2008-04699-C03-03) from the Spanish MICINN. G. Dalzotto University of Pisa, Largo Pontecorvo 5, Pisa, Italy T. Recio (B ) Universidad de Cantabria, Avda. Los Castros, Santander, Spain e-mail: tomas.recio@unican.es