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 6 of 6: Verify the converse

Previous step Next step

In plain words

Regular polygons have all chord-bisector reflection symmetries.

S=\{\text{vertices of a regular }r\text{-gon}\},\quad r\ge3

Detailed analysis

For the vertices of any regular polygon, the perpendicular bisector of every chord is a reflection axis of the polygon, hence of its vertex set. Therefore exactly the vertex sets of regular polygons satisfy the condition.