莫利四面体的两个逆猜想反例及一个新猜想
The Converse Problem for the Morley Tetrahedron: Counterexamples, Conjectures, and Partial Results
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中文总结 AI 辅助
本文给出莫利四面体两个逆猜想的反例，提出新猜想并证明其在多种特殊情形下成立，但一般情形仍开放。
中文摘要 AI 辅助
在先前的一篇论文中，作者证明了等腰四面体的莫利四面体仍然是等腰的，并提出了两个逆猜想。我们给出了这两个猜想的反例，并提出了一个新猜想：如果一个莫利四面体是正则的，那么重新标记后，原四面体有四条相等的交叉边。我们通过使用三次方程的简短论证证明了这一猜想对等腰四面体成立。我们还证明了当两对相对边相等、当四面体具有交换两个顶点的反射、或者当四个莫利顶点到其对应面的距离相等时，该猜想也成立。最后，我们证明了在三个已知例子附近不存在其他解。新模型GPT 6 Astra被用于尝试证明猜想4以及通过添加AB=CD得到的较弱猜想。两次尝试均未给出完整证明，两个猜想仍然开放。
英文摘要
In a paper in Acta Mathematica Hungarica the author proved that the Morley tetrahedron of an isosceles tetrahedron, obtained by trisecting the six dihedral angles, is again isosceles, and proposed two converse conjectures. We show that both are false. There is a nonisosceles tetrahedron $T_1$ and an isosceles, nonregular tetrahedron $T_2$ whose Morley tetrahedra are regular, and there are nonisosceles tetrahedra, even a two-parameter family of tetrahedra without any symmetry, whose Morley tetrahedra are isosceles. In $T_1$ and in $T_2$ there is a pair of opposite edges such that the other four edges are equal, and we conjecture that a regular Morley tetrahedron always forces this. We prove the conjecture for every tetrahedron with a nontrivial symmetry, and we show that, up to similarity, the regular tetrahedron, $T_1$ and $T_2$ are the only tetrahedra with this edge pattern and a regular Morley tetrahedron. The tetrahedron $T_2$ has $AB=CD=1$ and $AC=AD=BC=BD=\sqrt{(21+4\sqrt6)/45}$, while $T_1$ is given by a root of a sextic with Galois group $S_6$ and cannot be expressed by radicals. We also prove that a tetrahedron with a regular Morley tetrahedron is regular if it is orthocentric, if it is isodynamic, if its three sums of opposite edges are equal, if it has three equal edges at a vertex, if it has an equilateral face, or if none of its dihedral angles is larger than $95^\circ$; the tetrahedron $T_2$ has two dihedral angles of about $98.7^\circ$. For isosceles Morley tetrahedra we conjecture that $(AB^2-CD^2)(AC^2-BD^2)(AD^2-BC^2)\ge0$ and that $AB=CD$ forces a second pair of equal opposite edges. We also conjecture that a tetrahedron with $AC=BD$ whose Morley tetrahedron satisfies $A'B'=B'C'=C'D'=D'A'$ has a nontrivial symmetry. Some proofs are computer assisted; they use exact rational arithmetic or interval arithmetic with outward rounding.
