1999 IMO Problems/Problem 1

Problem

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 $B$ in $S$ , the perpendicular bisector of the line segment $AB$ is an axis of symmetry of $S$ .

Solution

Upon reading this problem and drawing some points, one quickly realizes that the set $S$ consists of all the vertices of any regular polygon.

Now to prove it with some numbers:

Let $S=\left\{ P_{0},P_{1},P_{2},...,P_{n-1} \right\}$ , with $n\ge 3$ , where $P_{i}$ is a vertex of a polygon which we can define their $xy$ coordinates as: $P_{i}=\left\langle Rcos\left( \frac{2\pi}{n}i \right),Rsin\left( \frac{2\pi}{n}i \right) \right\rangle$ for $i=0,1,...,(n-1)$ That should define the vertices of any regular polygon with $R$ being the radius of the circumscircle of the polygon.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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1999 IMO Problems/Problem 1

Problem

Solution