The many ways to construct a triangle and additional triangle facts . The many ways to construct a triangle Proposition I.1 of Euclid 's Elements deals with the construction of an equilateral triangle. Propositions I.4, I.8, and I.26 are what we nowadays would call SAS, SSS, ASA theorems, respectively.
Triangle is the most basic, simplest of all geometric shapes.
It may be argued that circle, not having corners and needing only one quantity to be well defined, may be simpler. Still it's the simplest among all polygons, and, I would speculate that, among all the simplest shapes, triangle offers the greatest variety of forms and definitions. A triangle is of course well defined by its vertices. As a set, they may be rotated, translated or reflected - the triangle will remain the same. In general, a triangle is defined by its three elements.