A set
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
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](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
from
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
, the perpendicular bisector plane of the segment
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
is a plane of symmetry for
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. 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.
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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.