A note on some generalizations of Monge's theorem

We generalize Monge's theorem for $n+1$ pairwise homothetic sets (in particular convex bodies) in $E^n$ in place of three disks in $E^2$. We also present a version for $n+1$ independent points of $E^n$. It also includes the reverse statement. Moreover, we give an analogon of Monge's theorem for the $n$-dimensional sphere and hyperboloid model of the hyperbolic space.