Problem 2

Given a point

P_0

in the plane of triangle

A_1A_2A_3

, define

A_s=A_{s-3}

for all

s\ge4

. Construct points

P_1,P_2,\ldots

so that

P_{k+1}

is the image of

P_k

under a clockwise rotation through

120^\circ

about

A_{k+1}

. Prove that if

P_{1986}=P_0

, then triangle

A_1A_2A_3

is equilateral.

Step 5 of 5: Conclude equilateral

In plain words

The point $a$ is exactly the third vertex of a unit equilateral triangle on the segment from $0$ to $1$ .

|A_1A_2|=|A_2A_3|=|A_3A_1|=1

Detailed analysis

Thus $A_1=0$ , $A_2=1$ , and $A_3=\frac12+i\frac{\sqrt3}{2}$ are the vertices of an equilateral triangle, completing the proof.