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 2 of 6: Show the hull is equilateral

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In plain words

Reflect the two neighboring hull vertices.

A_iA_{i+1}=A_{i+1}A_{i+2}

Detailed analysis

The perpendicular bisector of $A_iA_{i+2}$ is an axis of symmetry. Since $A_{i+1}$ is the intervening hull vertex, it must lie on this axis; hence $A_iA_{i+1}=A_{i+1}A_{i+2}$ . Repeating around the hull makes all sides equal.