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 classification of arithmetic triangle groups (there are only finitely many) and determined when two of them are commensurable. The particular example is related to Bring's curve and the arithmetic theory implies that the triangle group for Δ contains the triangle group for the triangle Δ with angles /4, /4 and /5 as a non-normal subgroup of index 6. Doubling the triangles Δ and Δ, this implies that there should be a relation between 6 triangles Δ with angles /2, /5 and /5 and hyperbolic area /10 and a triangle Δ with angles /5, /5 and /10 and hyperbolic area 3/5.