Difference between revisions of "1986 IMO Problems/Problem 2"
(New page: Given a point <math>P_0</math> in the plane of the triangle <math>A_1A_2A_3</math>. Define <math>A_s=A_{s-3}</math> for all <math>s\ge4</math>. Construct a set of points <math>P_1,P_2,P_3,...) |
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Revision as of 17:54, 9 September 2008
Given a point in the plane of the triangle . Define for all . Construct a set of points such that is the image of under a rotation center through an angle clockwise for . Prove that if , then the triangle is equilateral.
Solution
Consider the triangle and the points on the complex plane. Without loss of generality, let , , and for some complex number . Then, a rotation about of sends point to point . For , the rotation sends to and for the rotation sends to . Thus the result of all three rotations sends to
Since the transformation occurs times, to obtain . But, we have and so we have
Now it is clear that the triangle is equilateral.