Re: Some theorems about polyhedra


Dr Tim wrote:
1. If five faces of a distorted cube are cyclic quadrilaterals, then
the sixth is too.
2. If five faces of a distorted cube are quadrilaterals-of-tangents
(QOTs), then the sixth is too,
3. If all-but-one of the faces of a trihedral polyhedron are cyclic,
then the last is too.
4. If all-but-one of the faces of a polyhedron-with-quadrilateral-
faces is a QOT, then the last is too.

Does anyone recognise these theorems?
If not, you saw them here first!

Tim Robinson

Dear colleague,
the first property is the same as a Miquel theorem (in the plane)
if each of the five 4-uplets of points ABCD , ABA'B' , BCB'C' , CDC'D'
and ADA'D' is cocylic, then the sixth one A'B'C'D' is also cocyclic.

An inversion sending the sphere with your eight points to a plane
relates the properties.

Sincerely

.

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