1977 IMO Problems/Problem 1
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Problem
In the interior of a square we construct the equilateral triangles
Prove that the midpoints of the four segments
and the midpoints of the eight segments
are the 12 vertices of a regular dodecagon.
Solution
Just use complex numbers, with ,
,
and
.
With some calculations, we have
,
,
and
.
Now it's an easy job to calculate the twelve midpoints and to find out they are all of the form
, with
, and the result follows.
The above solution was posted and copyrighted by Joao Pedro Santos. The original thread for this problem can be found here: [1]
See Also
1977 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |