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Let two distinct $N$-simplexes be given in an Euclidean or pseudo-Euclidean$N+1$ dimensional space as each is defined by the coordinates of its $N+1$vertexes. We consider the two families of $N$-spheres passing through thevertexes of the given $N$-simplexes and the set of couples of $N$-spheres (onebelonging to first family and the other to the second one). The elements ofthis set have at least one common point; moreover, it is such that for theangle $\alpha$ between the segments connecting that point and the centers ofthe corresponding $N$-spheres, there holds $\cos^{2}\alpha=const$ for each ofthe elements of the defined set of $N$-spheres. In the present work we find thegeometric place of all these common points, including the special cases when$\cos^{2}\alpha$ is equal to $0$ or $1$.

On the common points of two families of $N$-spheres in the flat $N+1$  dimensional space, each of which passes through the vertexes of a given  $N$-simplex