1999 IMO Problems/Problem 1: Difference between revisions
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Let point <math>M</math>, not in <math>S</math> be a point that passes through the perpendicular bisector of <math>AB</math> at a distance <math>R</math> from <math>O</math> | Let point <math>M</math>, not in <math>S</math> be a point that passes through the perpendicular bisector of <math>AB</math> at a distance <math>R</math> from <math>O</math> | ||
Then, <math>\angle P_{0}OM =\frac{2\pi}{n}\frac{a+b}{2}</math> | |||
{{alternate solutions}} | {{alternate solutions}} | ||
Revision as of 19:49, 12 November 2023
Problem
Determine all finite sets 
of at least three points in the plane which satisfy the following condition:
For any two distinct points 
and 
in , the perpendicular bisector of the line segment 
is an axis of symmetry of .
Solution
Upon reading this problem and drawing some points, one quickly realizes that the set consists of all the vertices of any regular polygon.
Now to prove it with some numbers:
Let 
, with 
, where 
is a vertex of a polygon which we can define their 
coordinates as: 
for 
.
That defines the vertices of any regular polygon with 
being the radius of the circumcircle of the regular 
-sided polygon.
Now we can pick any points and of the set as:
and 
, where 
; 
; and 
Then,
and 
Let 
be point 
which is not part of
Then, 
, and 
The perpendicular bisector of passes through .
Let point 
, not in be a point that passes through the perpendicular bisector of at a distance from
Then, 
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
