Side Lengths of Morley Triangles and Tetrahedra

The famous Morley theorem says that the adjacent angle trisectors of a triangle form an equilateral triangle. We recall some known proofs and provide several new. In hyperbolic geometry, we compute the side lengths of the associated Morley triangle and show that the limit is the Euclidean (flat) equilateral case. The perspectivity properties hold also in general. We introduce new invariants such as the Morley polynomial and Morley group. Finally, we consider the space analogue and compute the side lengths of the Morley tetrahedron.