TITLE: Can Morley's theorem be generalized?
QUESTION [26 upvotes]: Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
In a talk some years ago, David Rusin made the provocative claim that Morley's theorem is a rare example of a striking theorem that defies generalization.  The first ideas that come to everyone's mind—passing to higher dimensions or hyperbolic geometry for example—don't work.
The proof by Alain Connes yields a mild generalization of sorts, but not a very satisfying one in my opinion.  Wikipedia claims that there are "various generalizations" of Morley's theorem, but by this it seems to mean extensions of Morley's theorem, i.e., further equilateral triangles that one can construct.  This is not what I would, strictly speaking, call a "generalization."
So is David Rusin correct?

Are there no satisfactory generalizations of Morley's theorem?

REPLY [2 votes]: A recent "generalization":

Tran, Q. H. "Morley’s trisector Theorem for isosceles tetrahedron." Acta Mathematica Hungarica (2021): 1-8.
DOI.

Abstract.
We extend Morley’s trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space. We will show that the Morley tetrahedron of an isosceles tetrahedron is also isosceles tetrahedron. Furthermore, by the formula for distance in barycentric coordinate, we introduce and prove a general theorem on an isosceles tetrahedron.
A "Morley tetrahedron" is determined by planes trisecting each
dihedral angle.
Tran, Q. H. is likely MO user TranQuangHung.