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 1 of 5: Normalize and encode a rotation

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In plain words

A similarity of the plane lets us place the first two centers at $0$ and $1$ ; a clockwise $120^\circ$ rotation is multiplication by the unit complex number $\omega$ .

A_1=0,\quad A_2=1,\quad A_3=a,\quad \omega=e^{4i\pi/3}

The clockwise

120^\circ

multiplier

\omega=e^{4i\pi/3}

on the unit circle.

Detailed analysis

Put $A_1=0$ , $A_2=1$ , $A_3=a$ and $\omega=e^{4i\pi/3}$ . A rotation about a center $c$ sends $P$ to $\omega(P-c)+c$ .