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 4 of 5: Solve for the third vertex

In plain words

The zero translation condition determines the location of $A_3$ relative to $A_1A_2$ exactly.

a=\frac{i\sqrt3}{1-e^{4i\pi/3}}=\frac12+i\frac{\sqrt3}{2}

Detailed analysis

Rearranging gives $a(1-\omega)=i\sqrt3$ . With $\omega=e^{4i\pi/3}$ , $a=\frac{i\sqrt3}{1-e^{4i\pi/3}}=\frac12+i\frac{\sqrt3}{2}$ .