This is an experimental geometry. All Hilbert's 20 axioms of the Euclidean GGeometry are denied in this vanguardist geometry of the real chaos: What is even more intriguing? F.Smarandache[5] has even found in 1969 a model of it:

This chapter briefly describes the computer proof system CDEG, version 2.0, a computerized formal system for giving diagrammatic proofs in Euclidean Geometry. The existence of such a computer proof system shows that geometric arguments that rely on diagrams can be made rigorous, and this fact has important philosophical implications for how we understand and teach geometry. The full details of ...

We prove the result from the title using the geometry of Euclidean buildings.

We collect dozens of well-known and not so well-known fundamental unsolved problems involving low dimensional submanifolds of Euclidean space. The list includes selections from differential geometry, Riemannian geometry, metric geometry, discrete or polyhedral geometry, geometric knot theory, theory of convex bodies, and integral geometry. The common thread through these selections are the simp...

In [Comput. Math. Appl. 41 (2001), 135–147], A.A. Ungar employs the Möbius gyrovector spaces for the introduction of the hyperbolic trigonometry. This Ungar’s work plays a major role in translating some theorems from Euclidean geometry to corresponding theorems in hyperbolic geometry. In this paper we explore the theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry.