Školjka

%V0 A set $S$ of points from the space will be called completely symmetric if it has at least three elements and fulfills the condition that for every two distinct points $A$ and $B$ from $S$, the perpendicular bisector plane of the segment $AB$ is a plane of symmetry for $S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.