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
Schwarz triangle function hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles without "cusps"—angles equal to zero or equivalently vertices on the real axis—the elementary approach of will be followed. For triangles with one or two cusps, elementary arguments of , simplifying the approach of , will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a Hecke group. For an ideal triangle in which all angles are zero, so that all vertices lie on the real