It is only required that at least two sides be equal in order for a triangle to be isosceles.Įquilateral triangles are constructed in the very first proposition of the Elements, I.1. The way that the term isosceles triangle is used in the Elements does not exclude equilateral triangles. The term isosceles triangle is first used in proposition I.5 and later in Books II and IV. The equilateral triangle A not only has three bilateral symmetries, but also has 120°-rotational symmetries.Īccording to this definition, an equilateral triangle is not to be considered as an isosceles triangle. The scalene triangle C has no symmetries, but the isosceles triangle B has a bilateral symmetry. This definition classifies triangles by their symmetries, while definition 21 classifies them by the kinds of angles they contain. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
