1986 IMO Problems/Problem 2
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.