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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Now it is clear that the triangle <math>A_1A_2A_3</math> is equilateral. | Now it is clear that the triangle <math>A_1A_2A_3</math> is equilateral. | ||
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+ | == See Also == {{IMO box|year=1986|num-b=1|num-a=3}} |
Revision as of 23:04, 29 January 2021
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.
See Also
1986 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |