The paper gives a formulation of Thales' theorem that enables the definition of segment multiplication by avoiding the explicit connection to parallelism. This simplifies the approach to segment multiplication given in Hilbert's Grundlagen der Geometrie, which is based on Pascal's theorem. Moreover, it suggests an idea of equiangularity that provides a more general formulation of many theorems independently of the different Euclidean or non-Euclidean settings. The relationship of Thales' theorem with other important theorems, and with incommensurability, is considered, which illuminates its cruciality in the organization of Euclid's Elements.
Vincenzo Manca (Wed,) studied this question.