A multidimensional analog of the Conway circle

Conway established the following geometric fact: If the sides $AB$ and $AC$ of a triangle $ABC$ are prolonged beyond the point $A$ by the length of the opposite side $BC$ and the same is done with the vertices $B$ and $C$, then the so-constructed $6$ points lie on the sole circle whose center coincides with the center of the inscribed circle. V.A. Alexandrov found a spatial analog of the Conway circle. Namely, if in a tetrahedron $ABCD$ we mark three points on the prolongations of the edges $AB$, $AC$, and $AD$ beyond the vertex $A$ at distance from $A$ to the half-perimeter of the opposite face $BCD$ and then do the same with the remaining vertices $B$, $C$, and $D$ then the so-constructed $12$ points lie on the same sphere if and only if $ABCD$ is a frame tetrahedron. We address the multidimensional version of the fact for a simplex in the Euclidean space $E_n$.