Geometric Loci and ChatGPT: Caveat Emptor!

Abstract

We compare the performance of two systems, ChatGPT 3.5 and GeoGebra 5, in a restricted, but quite relevant, benchmark from the realm of classical geometry: the determination of geometric loci, focusing, in particular, on the computation of envelopes of families of plane curves. In order to study the loci calculation abilities of ChatGPT, we begin by entering an informal description of a geometric construction involving a locus or an envelope and then we ask ChatGPT to compute its equation. The chatbot fails in most situations, showing that it is not mature enough to deal with the subject. Then, the same constructions are also approached through the automated reasoning tools implemented in the dynamic geometry program, GeoGebra Discovery, which successfully resolves most of them. Furthermore, although ChatGPT is able to write general computer code, it cannot currently output that of GeoGebra. Thus, we consider describing a simple method for ChatGPT to generate GeoGebra constructions. Finally, in case GeoGebra fails, or gives an incorrect solution, we refer to the need for improved computer algebra algorithms to solve the loci/envelope constructions. Other than exhibiting the current problematic performance of the involved programs in this geometric context, our comparison aims to show the relevance and benefits of analyzing the interaction between them.