Integer triangle common divisor. Integer triangles with a 60° angle can also be generated by with coprime integers "m", "n" with 0 < "n" < "m" (the angle of 60° is opposite to the side of length "a"). From here, all primitive solutions can be obtained by dividing "a", "b", and "c" by their greatest common divisor (e.g. an equilateral triangle solution is obtained by taking "m" = 2 and "n" = 1, but this produces "a" = "b" = "c" = 3, which is not a primitive solution). See also More precisely, If formula_109, then formula_110, otherwise formula_111. Two different pairs
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