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
Triangle β is the interior angle at "B", formula_28 is the interior angle at "C" and "c" is the line AB). Furthermore, since sin α = sin ("π" − α) = sin (β + formula_28), and similarly for the other two angles: "Knowing AAS": and analogously if the known side is "a" or "c". "Knowing ASA": and analogously if the known side is "b" or "c". The shape of the triangle is determined by the lengths of the sides. Therefore, the area can also be derived from the lengths of the sides. By Heron's formula: where formula_34 is the semiperimeter, or