Hyperbolic and spherical power of a circle

Suppose that a line passing through a given point P intersects a given circle C at Q and R in the Euclidean plane. It is well known that |PQ||PR| is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.