Higher-dimensional Extension of Triangle Geometry? 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
$$ \sum_{1\le i,j\le n} {D}_{ij}x_ix_j=0$$
 A: 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.
