# When do $n+2$ points in $\mathbb{R}^n$ lie on a same $(n-1)$-sphere?

When $$n=2$$, the following results are well-known:

Proposition 1. Let $$A,B,C,D$$ be $$4$$ distinct points in $$\mathbb{R}^2$$. They are aligned or cocyclic if and only if: $$\left(\overrightarrow{CA},\overrightarrow{CB}\right)\equiv\left(\overrightarrow{DA},\overrightarrow{DB}\right)\mod \pi.$$

Proposition 2. (Ptolemy's theorem) Let $$A,B,C,D$$ be $$4$$ distinct points in $$\mathbb{R}^2$$. They are cocyclic if and only if one of the following equalities holds true: $$AB.CD\pm AC.DB\pm AD.BC=0.$$

In this recent question it is proven that whenever $$n+1$$ points in $$\mathbb{R}^n$$ do not lie in any affine hyperplane, they are on a unique $$(n-1)$$-sphere, which leads me to ask the following:

Question. Is there a necessary and sufficient condition to determine when $$n+2$$ points in $$\mathbb{R}^n$$ are on a same affine hyperplane or on a same hypersphere?

I easily derived from the equations of an affine hyperplane and of an hypersphere that if $$x_i:=(x_{i,j})$$ are $$n+2$$ points of $$\mathbb{R}^n$$, the $$x_i$$s are on a same hyperplane or lie on a same hypersphere if and only if: $$\left|\begin{matrix}{x_{1,1}}^2+\cdots+{x_{1,n}}^2&x_{1,1}&\cdots&x_{1,n}&1\\{x_{2,n}}^2+\cdots+{x_{2,n}}^2&x_{2,1}&\cdots&x_{2,n}&1\\\vdots&\vdots&\ddots&\vdots&\vdots\\{x_{n+1,1}}^2+\cdots+{x_{n+1,n}}^2&x_{n+1,1}&\cdots&x_{n+1,n}&1\\{x_{n+2,1}}^2+\cdots+{x_{n+2,n}}^2&x_{n+2,1}&\cdots&x_{n+2,n}&1\end{matrix}\right|=0.$$ However, I am more interested in a characterization involving angles in the same way as in proposition 1. or distances like in proposition 2. In particular, in the case $$n=3$$ is there a necessary and sufficient condition expressing a relation between solid angles?

Regarding the case $$n=3$$, my guess would be to determine the set of points from where one can observe a given circle with a constant solid angle.

Assume $x_0 = 0$ for simplicity and let $x_i' = \frac{x_i}{|x_i|^2}$ be the images of $x_i$'s under an inversion centered at $x_0$. By a well-known property of inversions, $x_0,\ldots,x_{n+1}$ lie on an affine $n-1$-plane or an $n-1$-sphere if and only if $x_1',\ldots,x_{n+1}'$ lie on an affine $n-1$-plane.