Forget that stuff about uniqueness and avoiding intersections --- the only constraint missing was bijection between 'scribed points and original facets. WFL On 2/8/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
My definitions should have specified that in/exscribed vertices correspond bijectively and uniquely with original facets which they meet.
Dan's 2-D picture fails to make the cut, since two vertices lie on line F_0 . As do tricks such as placing scribed vertices at the intersections of original [ (n-1)-dimensional ] facets.
As for the situation in n = 2 dimensions ... good question!
WFL
On 2/8/16, Dan Asimov <dasimov@earthlink.net> wrote:
On Feb 7, 2016, at 6:13 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The following problem (or rather its solution) constitutes the crux of what appears to be a new result in elementary 3-space Euclidean geometry. However its solution proves so neat that I'm posing it on the list, ahead of the solution to follow in a few days. [A handy excuse for further procrastination over the next instalment of Poncelet pontification.]
Gloss (0) : Given an arbitrary tetrahedron A , denote by B the inscribed tetrahedron with vertices the centroids of faces of A . It is well known that B is similar to A , but scaled down to 1/3 ; and an analogous result holds for any dimension n , with scale-down 1/n .
Given A and a chosen vertex A_0 , a tetrahedron is `exscribed' when its vertices meet the extended face planes of A , but lie outside face plane F_0 opposite A_0 .
I don't think I'm understanding this definition, or how the Dropbox picture relates to it.
Does it have an analogue for triangles in 2D ?
Would that be something like this, where the *'s are the vertices of the original triangle and the o's of the exscribed one with respect to A0:
A0 | *
F0-----*----o---*----
o o
???
—Dan
Problem (1) : Construct explicitly some exscribed C congruent to A . For example, grey A and yellow C in diagram https://www.dropbox.com/s/blcihoao8t0k3lh/exscribe2.png
Problem (2) : Generalise the result to a simplex in Euclidean n-space, for n >= 3 .
Question (3) : Investigate whether C is the smallest possible exscribed on F_0 and similar to A . (Probably, but I don't know either.)
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