Equilateral triangle Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Denoting the common length of the sides of the equilateral triangle as "a", we can determine using the Pythagorean theorem that: Denoting the radius of the circumscribed circle as "R", we can determine using trigonometry that: Many of these quantities
Weitzenböck's inequality that the sum of areas of the equilateral triangles is always greater or equal than the threefold area of the original triangle. This can now can be shown by replicating area of the triangle three times within the equilateral triangles. To achieve that the Fermat point is used to partition the triangle into three obtuse subtriangles with a formula_8 angle and each of those subtriangles is replicated three times within the equilateral triangle next to it. This only works if every angle of the triangle is smaller than formula_8, since otherwise the Fermat point is not located in the interior