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 5 of 6: Exclude interior points

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In plain words

An interior point cannot lie on the hull’s circumcircle.

OA_i=OA_{i+1}=OX

Detailed analysis

The perpendicular bisectors of $A_iX$ and $A_{i+1}X$ are also symmetry axes, so both pass through $O$ . Hence $OA_i=OX=OA_{i+1}$ . But every point strictly inside triangle $OA_iA_{i+1}$ lies strictly inside the circumcircle through $O,A_i,A_{i+1}$ , whereas $X$ lies on that circle, a contradiction. Therefore $S$ has no extra points.