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 3 of 5: Use the periodicity

In plain words

After each block, the point receives the same translation, so after $662$ blocks the displacement is $662$ times that vector.

1986=3\cdot662

Detailed analysis

Because $1986=3\cdot662$ , $P_{1986}=P_0+662(-i\sqrt3-a\omega+a)$ . The hypothesis $P_{1986}=P_0$ therefore gives $-i\sqrt3-a\omega+a=0$ .