Morley's trisector theorem for isosceles tetrahedron (Q2064769)

The author continues the work of \textit{D. Svrtan} and \textit{D. Veljan} [Forum Geom. 17, 123--142 (2017; Zbl 1366.51005)] on the Morley tetrahedron of a given tetrahedron. \par Let \(1 < p \in \mathbb{R}\). The author calls \(E\) a \(p\)-plane iff the angle between two faces of a given tetrahedron is \(p\) times the angle between \(E\) and one of the faces. For \(2\neq p\), there exist two \(p\)-planes containing a fixed edge, \(p=3\) gives the trisecting situation. \par Now, the author defines any of the four edges of the (generalized) \textit{Morley tetrahedron} of a given tetrahedron by intersecting three \(p\)-planes each of which is chosen with three arbitrary real parameters \(e,f,g > 1\). The main result of the paper under review is: For any isosceles tetrahedron the generalized Morley tetrahedron must be isosceles too. With \(e=f=g=3\) one gets the well-known Morley situation in dimension three. The proof uses homogeneous barycentric coordinates to calculate lengths of opposite edges.