Problem 1

Determine all finite sets

S

of at least three points in the plane which satisfy the following condition: For any two distinct points

A

and

B

S

, the perpendicular bisector of the line segment

AB

is an axis of symmetry of

S

Step 4 of 6: Fix the center

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In plain words

All symmetries of the hull pass through its center.

O\text{ is the center of the regular hull}

Detailed analysis

Every symmetry axis of $S$ preserves its convex hull and therefore fixes the center $O$ of the regular polygon. If $X\in S$ were not a hull vertex, choose a triangle $OA_iA_{i+1}$ containing it.