Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane

The hyperbolic cosines and sines theorems for the curvilinear triangle bounded by circular arcs of three intersecting circles are formulated and proved by using the general complex calculus. The method is based on a key formula establishing a relationship between exponential function and the cross-ratio. The proofs are carried out on Euclidean plane.

A new proof for the residual set dimension of the apollonian packing

Let (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.