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
Triangle of his "Elements", around 300 BC. The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An "exterior angle" of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to