Law of cosines . The power of the point with respect to the circle is equal to both and . Therefore, which is the law of cosines. Using algebraic measures for line segments (allowing negative numbers as lengths of segments) the case of obtuse angle () and acute angle () can be treated simultaneously. By using the law of sines and knowing that the angles of a triangle must sum to 180 degrees, we have the following system of equations (the three unknowns are the angles): Then, by using the third equation of the system, we obtain a system of two equations in
Schwarz triangle function since the case of two cusps is obtained by doubling a triangle with one cusp and non-zero angles /"a" and /2. So consider the geodesic triangle in the upper half plane with angles 0, /"a", /"b" with "a", "b" integers greater than one. The interior of such a triangle can be realised as the region "X" in the upper half plane lying outside the unit disk |"z"| ≤ 1 and between two lines parallel to the imaginary axis through points "u" and "v" on the unit circle. Let Γ be the triangle group generated by the three reflections in the