On preserving the orientation of triangle under quasi-isometric mapping

In the article the sufcient sign of preserving the orientation of a triangle under quasi-isometric mapping is formulated and proved. The received result can be considered as synthesis of the Alfors' theorem on preserving the orientation of the exact triangle under quasiconformal mapping. The result is formulated for the arbitrariest triangle. It is shown that for an equilateral triangle, assessment characteristics of mapping are weaker than in the specifed theorem. The proof is based on application of the concept of distance between families of points, discussed by us earlier.