Generalizing the Steiner–Lehmus theorem using the Gröbner cover

Abstract

In this note we present an application of a new tool (the Gröbner cover method, to discuss parametric polynomial systems of equations) in the realm of automatic discovery of theorems in elementary geometry. Namely, we describe, through a relevant example, how the Gröbner cover algorithm is particularly well suited to obtain the missing hypotheses for a given geometric statement to hold true. We deal with the following problem: to describe the triangles that have at least two bisectors of equal length. The case of two inner bisectors is the well known, XIX century old, Steiner–Lehmus theorem, but the general case of inner and outer bisectors has been only recently addressed. We show how the Gröbner cover method automatically provides, while yielding more insight than through any other method, the conditions for a triangle to have two equal bisectors of whatever kind.