Counterexamples to Two Converse Conjectures for the Morley Tetrahedron and a New Conjecture
In an earlier paper, the author proved that the Morley tetrahedron of an isosceles tetrahedron is again isosceles and proposed two converse conjectures. We give counterexamples to both and propose a new conjecture: if a Morley tetrahedron is regular, then the original tetrahedron has four equal cross edges after relabelling. We prove this for isosceles tetrahedra by a short argument using a cubic equation. We also prove it when two pairs of opposite edges are equal, when the tetrahedron has a reflection interchanging two vertices, or when the four Morley vertices have equal distances from their corresponding faces. Finally, we show that no other solutions lie sufficiently close to the three known examples. The new model GPT 6 Astra was used to attempt proofs of Conjecture~\ref{conj:four} and of the weaker conjecture obtained by adding $AB=CD$. Neither attempt gave a complete proof, and both conjectures remain open.