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 1 of 6: Pass to the convex hull

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In plain words

The extreme points control every symmetry of the finite set.

A_1,\ldots,A_r

Detailed analysis

Let $A_1,\ldots,A_r$ be the vertices of the convex hull of $S$ , in cyclic order. The symmetry condition for every pair applies in particular to hull vertices.