The Hyperbolic Pythagorean Theorem in the Poincare Disc Model of Hyperbolic Geometry

Sometime in the sixth century B.C. Pythagoras of Samos discovered the theorem that now bears his name in Euclidean geometry. The extension of the Euclidean Pythagorean theorem to hyperbolic geometry, which is commonly known as the hyperbolic Pythagorean theorem (see [3, 5, 6, 9-11]), does not have a form analogous to the Euclidean Pythagorean theorem, so some authors have concluded that a truly hyperbolic Pythagorean theorem does not exist. For example, Wallace and West assert "the Pythagorean theorem is strictly Euclidean" since "in the hyperbolic [Poincare disc] model the Pythagorean theorem is not valid!" [15]. We show that a natural formulation of the hyperbolic Pythagorean theorem does exist: it expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as a natural "sum" of the squares of the hyperbolic lengths of the other two sides. The most general M6bius transformation of the complex unit disc D= {z: lzl < 1} in the complex z-plane [2,4,8],