scalene, isosceles and equilateral triangles

One natural way to sort triangles is to count how many of their three sides are equal. A scalene triangle has all three sides of different lengths — no two match. An isosceles triangle has at least two sides equal. An equilateral triangle has all three sides equal, the most symmetric of the three.

Precisely: in triangle ABC, if |AB|, |BC|, |CA| are all distinct it is scalene; if exactly two (or more) are equal it is isosceles; if all three are equal it is equilateral. Note that equilateral is a special case of isosceles under the usual 'at least two' definition, so every equilateral triangle is also isosceles. The side classification quietly controls the angles too: by the isosceles-triangle theorem, equal sides force equal opposite angles, so an isosceles triangle has two equal base angles and an equilateral triangle has all three angles equal at 60 degrees each (an equiangular triangle).

A common slip is to treat 'isosceles' as meaning exactly two equal sides; under the standard definition it means at least two, so the equilateral triangle is included. Also keep this side-based scheme separate from the angle-based one (acute, right, obtuse): a triangle has one label from each list, for example 'isosceles right triangle'.