Let $S \subset \mathbb{R}^n$ be a $n$-simplex. Let $a_0,\dots, a_n$ be the vertices of $S$. Define $L_i\subset \mathbb{R}^n$ be the hyperplane which touches $S$ at $a_i$ and parallel to the convex hull of $\{a_1,\dots,a_n\}\setminus \{a_i\}$.
How can I see that he region $B$ bounded by $L_0,\dots,L_n$ is a simplex similar to $S$?
Let me try to expand (and fix!) my comment into an answer:
If $a=a_0+\cdots+a_n$, I claim that the similarity $$ \begin{array}{rccc} F:&\mathbb{R}^n&\longrightarrow&\mathbb{R}^n\\ &v&\longmapsto& -nv+a \end{array} $$ sends $S$ to $B$. To prove this we shall see that $F(a_i)$ lies in all the hyperplanes except for $L_i$, which means that $F(a_i)$ is a vertex of $B$.
Let $j\neq i$, is it true that $F(a_i)\in L_j$? This happens iff $F(a_i)-a_j$ is parallel to the hyperplane defined by $\{a_0,\ldots,a_n\}\setminus\{a_j\}$, which is the hyperplane generated by the $n-1$ vectors $$a_0-a_i,\ldots,\widehat{a_j-a_i},\ldots,a_n-a_i.$$ Adding all of them up gives $a-a_i-a_j-(n-1)a_i=-na_i+a-a_j=F(a_i)-a_j$.
