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I am currently exploring generalizations of triangle geometry to higher dimensions. I know that "important questions" of Euclidean geometry have been already addressed and is considered obsolete; most mathematicians are more curious about "curved spaces" where many "basic" questions still remain as mysteries. However, I think that there are some very fascinating results in this Euclidean geometry that are worth some attention (although they likely don't have big impact in other areas of mathematics.)

Anyway, I am in particular trying to generalize several common concepts in traditional triangle geometry like circumsphere/circumcenter (circumscribed sphere to a given simplex and its center), insphere/incenter (inscribed sphere to a given simplex and its center), orthocenter (intersection of orthogonal lines drawn from vertices of a given simplex to its opposite facet). However it is more than possible that what I am working on is tackled previously.

Here's where I need your help. Does anyone know any previous references to studies as such?

I'll also state two theorems I discovered on the way, which may be done before individually in other contexts. Note that Theorem 1 is a further simplified version of a question that was addressed on Math.SE previously.

Definition. Given $n$ vectors in $\mathbb{R}^{n-1}$ in general position: $\{\vec{a}_1, \cdots, \vec{a}_n\}$, any point $\vec p\in \mathbb{R}^{n-1}$ can be expressed as a normalized linear combination $\vec p =\sum p_i \vec a_i$ where $\sum p_i=1$. Then we denote $(p_1,\cdots,p_n)$ by $[\vec p]$ and we call it (absolute) barycentric coordinates of $\vec p$.

Theorem 1. If two points $\vec p,\vec q$ have barycentric coordinates $[\vec p]=(p_1,\cdots,p_n)$ and $[\vec q]=(q_1,\cdots, q_n)$, then we can express distance between the two points in terms of barycentric coordinates as follows: $$||\vec p -\vec q||^2= -\frac12\sum_{1\le i,j\le n} (p_i-q_i)(p_j-q_j)||\vec{a}_i-\vec{a}_j||^2$$

Theorem 2. Denote by $D$ the matrix such that $D_{ij}=||\vec a_i-\vec a_j||^2$. There exists a unique sphere $S$ for which $\{\vec{a}_1,\cdots,\vec{a}_n\}\subset S$. Then a point with barycentric coordinate $(x_1,\cdots, x_n)$ lies on this sphere iff $$ \sum_{1\le i,j\le n} {D}_{ij}x_ix_j=0$$

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"Given $n$ linearly independent vectors in $\mathbb{R}^{n-1}$" will lead you to be able to show anything at all, as there are not that many lin. independent vectors there. –  Tobias Kildetoft Aug 9 '12 at 7:53
sorry, that was a mistake. let me correct that –  progressiveforest Aug 9 '12 at 7:56
You might be interested in the field of Computational Geometry –  Nick Alger Aug 19 '12 at 21:25
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2 Answers

I'm sure others will come up with much more (I know next to nothing about non-trivial Euclidean geometry), but here's a book I've had for about 40 years that (as I just now learned) is freely available on the internet and could be of interest to you:

Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry by Frederick S. Woods (1922)

Although I don't really know the subject matter of Woods' book, over the years that I've had my copy, I've flipped through it quite a bit and I've read bits and pieces here and there in it, and it is on this basis that I think much of what you're asking about can be found in Woods' book (but in old fashioned language and in old fashioned terminology).

If you can read French, there is a huge amount of literature available on properties of tetrahedrons. For example, see the following memoir:

Mémoire sur le Tétraèdre by Joseph Neuberg (1886). Neuberg's memoir consists of the first 72 pages of this google-books item.

If you can read French and have access to a decent library (or are willing to google for freely available digitized volumes), the journal Mathesis Recueil Mathématique (the published volumes date from 1881 to 1965) has a large number of articles on properties of tetrahedrons other non-trivial topics in 2- and 3-dimensional Euclidean geometry.

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thanks for the reference! –  progressiveforest Aug 20 '12 at 17:14
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I have been interested in extending theorems of plane euclidean geometry to higher dimensions for some time, and I am willing to send pdf offprints of my work to those interested. Circumcenters, incenters, centroids of simplices are common material in textbooks on tetrahedra and higher dimensional simplices (see Court (English), Thebault (French), Couderc and Ballicioni (French). The orthocenter does not exist for all tetrahedra (i.e., altitudes are not always concurrent). Those simplices in which altitudes are concurrent are called orthocentric.

Together with others, I have several papers on orthocentric simlices, on the coincidence of centers, on higher dimensional versions of Propositions 5, 6 of Book 1 of Euclid's Elements (i.e., the pons asinorum or the bridge of asses), on higher dimensional versions of Propositions 24, 25 of Book 1 of Euclid's Elements (i.e., the open mouth theorem), on higher dimensional versions of Archimedes Arbelos theorem), and others.

If you are interested, please write to me for prints and exchanging thoughts.

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that sounds great! Could you tell me your email address for some discussion? –  progressiveforest Aug 24 '12 at 23:45
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