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\newcommand{\ds}{\displaystyle}
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\begin{document}


{\bf Question}

Show that a necessary and sufficient condition that  the points
denoted by $z_1,z_2$ and that the origin should form an
equilateral triangle is $$z_1^2 - z_1z_2 + z_2^2 = 0$$

\vspace{.25in}

{\bf Answer}

$O, z_1, z_2$ form an equilateral triangle if and only if $\ds z_2
= z_1e^{\pm i\frac{pi}{3}}$

if and only if $z_2^3 = -z_1^3$ and $z_2 \not= z_1$

if and only if $(z_1+z_2)(z_1^2 - z_1z_2 + z_2^2) = 0$ and $z_2
\not= z_1$

if and only if $(z_1^2 - z_1z_2 + z_2^2) = 0$

\end{document}
