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 2 of 5: Compose one block of three rotations

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In plain words

The same three centers repeat every three steps, so one block determines the entire long sequence.

P\longmapsto\omega\bigl(\omega(\omega P-1)+1-a\bigr)+a

Detailed analysis

Applying the rotations about $0$ , $1$ , and $a$ successively gives $P\mapsto\omega(\omega(\omega P-1)+1-a)+a$ . Since $\omega^3=1$ , this simplifies to $P\mapsto P-i\sqrt3-a\omega+a$ (with $\omega=e^{4i\pi/3}$ ).