Integer triangle formula_112 and formula_113 generate the same triple. Unfortunately the two pairs can both be of gcd=3, so we can't avoid duplicates by simply skipping that case. Instead, duplicates can be avoided by formula_27 going only till formula_115. Note that we still need to divide by 3 if gcd=3. The only solution for formula_116 under the above constraints is formula_117 for formula_118. With this additional formula_119 constraint all triples can be generated uniquely. An Eisenstein triple is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees. Integer triangles
Isosceles triangle to provide a solution. Although originally formulated only for internal angle bisectors, it works for many (but not all) cases when, instead, two external angle bisectors are equal. The 30-30-120 isosceles triangle makes a boundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external). The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles. The radius of the inscribed circle of an isosceles triangle with side length formula_2, base formula_3, and height formula_1 is: The center of the circle lies on the symmetry