  The purpose the present paper is to construct the hyperbolic trigonometry on
Euclidean plane without refereing to hyperbolic plane. In this paper we show
that the concept of hyperbolic angle and its functions forming the hyperbolic
trigonometry give arise on Euclidean plane in a natural way. The method is
based on a key- formula establishing a relationship between exponential
function and the ratio of two segments. This formula opens a straightforward
pathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic law
of cosines I and II and the hyperbolic law of sines are derived by using of the
key-formula and the methods of Euclidean Geometry, only. It is shown that these
laws are consequences of the interrelations between distances and radii of the
intersecting semi-circles.
