Integer triangle harmonic progression. This is because such angles have to be rational angles of the form with rational 0 < < 1. But all the angles of integer triangles must have rational cosines and this will occur only when = i.e. the integer triangle is equilateral. The square of each internal angle bisector of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle "A" is formula_2 where "s" is the semiperimeter (and likewise for the other angles' bisectors). Any altitude dropped from a vertex onto an opposite side or its extension will split
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