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.
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.