Finding the normal of a simplex facet in n-dimensions I am attempting to find a generalised formula for the normal of a simplex facet in n-dimensions.
For example if I had the 2 dimensional simplex formed by the vertices ABC below. Then I want to find the normal for the facet formed by the vertices BC. The reason I want the normal is because I'm trying to find the hyperplane that intersects vertices BC, and is equal to 1 at vertex A.
Is there a nice way of doing this?
One way I thought of would be to find D. and then use the formulas:
$$W = \frac{A-D}{||A-D||_2}$$
$$b = -D\cdot{W}$$
for the plane:
$$xW + b = y$$
However, I don't know how to find D for an n-dimensional simplex facet either.
I'm not very knowledgeable in linear algebra or geometry, so any help would be great.

 A: Let $x_0,\ldots,x_n$ be the vertices of the $n$-dimensional simplex.
You can find all normals of the facets using the inverse of the following matrix:
$$
\begin{pmatrix}
\;\; & x_0^T & \;\; & 1 \\
\;\; & x_1^T & \;\; & 1 \\
\;\; & \vdots & \;\; & \vdots \\
\;\; & x_n^T & \;\; & 1 \\
\end{pmatrix}
$$
Note that this matrix is invertible unless the simplex is degenerate. The determinant of this matrix is $\pm n!$ times the volume of the simplex.
Let's compose the inverse of the matrix of vectors $y_0,\ldots,y_n \in\mathbb{R}^n$ and numbers $c_0,\ldots,c_n\in\mathbb{R}$ as follows:
$$
\begin{pmatrix}
\;\; & x_0^T & \;\; & 1 \\
\;\; & x_1^T & \;\; & 1 \\
\;\; & \vdots & \;\; & \vdots \\
\;\; & x_n^T & \;\; & 1 \\
\end{pmatrix}
\begin{pmatrix}
 \; & & & \; \\
 y_0 & y_1 & \cdots & y_n \\
 \; & & & \; \\
-c_0 & -c_1 & \cdots & -c_n
\end{pmatrix}
=
\begin{pmatrix}
1 & & & 0 \\
 & 1 &  & \\
 & & \ddots & \\
0 &  &  & 1
\end{pmatrix}
$$
Then you get $\langle x_i\, ,\,y_i\rangle = 1+c_i$ and $\langle x_i\, ,\,y_k\rangle = c_k,\;i\neq k.$ This means
$$
i\neq k,\,j\neq k\; \Rightarrow \; \langle x_i-x_j\, , \,y_k \rangle 
=\langle x_i\, , \,y_k \rangle - \langle x_j\, , \,y_k \rangle
= c_k - c_k = 0
$$
which shows that $y_k$ is perpendicular to the facet opposite to $x_k.$
Note that the vectors $y_k$ are not necessarily unit vectors. You might have to scale them.
