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The purpose the present paper is to construct the hyperbolic trigonometry onEuclidean plane without refereing to hyperbolic plane. In this paper we showthat the concept of hyperbolic angle and its functions forming the hyperbolictrigonometry give arise on Euclidean plane in a natural way. The method isbased on a key- formula establishing a relationship between exponentialfunction and the ratio of two segments. This formula opens a straightforwardpathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic lawof cosines I and II and the hyperbolic law of sines are derived by using of thekey-formula and the methods of Euclidean Geometry, only. It is shown that theselaws are consequences of the interrelations between distances and radii of theintersecting semi-circles.

Hyperbolic cosines and sines theorems for the triangle formed by intersection of three semicircles on Euclidean plane