This condition: -----
And to fulfil the conditions of the problem, the triangles would have also to be similar ...
were never mentioned as part of the "problem", just as a condition that the centroid-tetrahedra (and higher dimensional analogues) satisfied, and one reason my 2D example failed. —Dan
On Feb 8, 2016, at 6:43 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I have redrawn the 3-D diagram, adding pink lines to join the vertices of each original (grey) face to the exscribed (yellow? khaki!) vertex which it meets: https://www.dropbox.com/s/wugu0q3ezrfkrdp/exscribe3.png Perhaps that will help clarify their relationship; or perhaps not.
Note that the bijection is natural, in the sense that each exscribed vertex lies in the plane of the face opposite to the original vertex to which it is congruent. Also the tetrahedra are mirror images, related by an `orientation reversing' isometry.
WFL
On 2/8/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
My mistake: your dots are original vertices, your "o"s are exscribed.
But for exscription the two lower "o"s should lie on the (extended) sides F_ 2 = A_0 A_1 and F_1 = A_0 A_2 of the dotted triangle.
And to fulfil the conditions of the problem, the triangles would have also to be similar ...
In my 3-D diagram, all the constraints are satisfied: each yellow vertex lies on just one grey face-plane, each grey plane meets one yellow vertex, and the tetrahedra are (in fact) congruent.
WFL
On 2/8/16, Dan Asimov <dasimov@earthlink.net> wrote:
Now I am more confused than ever.
The two vertices of the triangle I drew that are on the line F0 are the two vertices of the original triangle that are necessarily on the edge not containing the "chosen" vertex A0.
—Dan
On Feb 8, 2016, at 8:40 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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.)