On an argand diagram the points P and Q represent the numbers $\displaystyle z1$ and $\displaystyle z2$ respectively. OPQ is an equilateral triangle. Show that $\displaystyle (z1)^2 + (z2)^2 = z1z2$
Represent one number as $\displaystyle re^{i\varphi}$ and the other as $\displaystyle re^{i\varphi+i\pi/3}$ and use the fact that $\displaystyle e^{i\pi/3}=1+e^{i2\pi/3}$.