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 3 of 6: Show the hull is regular

Previous step Next step

In plain words

The same reflection argument equalizes the angles.

\angle A_iA_{i+1}A_{i+2}=\angle A_{i+1}A_{i+2}A_{i+3}

Detailed analysis

Applying the perpendicular-bisector symmetry to $A_i$ and $A_{i+3}$ shows the corresponding adjacent angles are equal. Thus all hull angles are equal as well as all side lengths, so the hull is a regular $r$ -gon.