On an Argand diagram, the points P and Q represent z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1² + z2² = z1z2
Since the sides of an equlateral triangle all have the same length sides we know:
|z1|=|z2|=|z1-z2|
The internal angles of the triangle are all 60 degress so we also know:
arg(z1) = arg(z2) + 60
And
$\displaystyle |z1^2|=|z2^2|$
And
arg(z1^2)=arg(z2^2) + 120
So if z1=$\displaystyle r\angle\theta$ then z2=$\displaystyle r\angle(\theta+60)$ and $\displaystyle z2^2=r^2\angle(2\theta+2*60)$
$\displaystyle z1^2+z2^2=r^2\angle2\theta + r^2\angle(2\theta+120)=r^2(1\angle2\theta+1\angle( 2\theta+120))=r^2(-1\angle(2\theta-120))$$\displaystyle =r^2(1\angle(2\theta+60))=r\angle\theta(r\angle(\t heta+60))=z1z2$