Question

$z_1,z_2$ and $z_3$ are the vertices of an isosceles triangle in anticlockwise direction with origin as in center, then show that $z_2,z_1,$ and $z_3$ are in G.P

Answer 1

Since Origin is the center, it is equidistant from all the three points.
And
$|z_2-z_3|=|z_2-z_1|$ ... isosceles triangle.
Hence
$\frac{z_3-z_0}{z_2-z_0}=\frac{z_2-z_0}{z_1-z_0}$
Now
$z_0=0$ ... origin.
Hence
$\frac{z_3}{z_2}=\frac{z_2}{z_1}$
$z_2^2=z_1{z}_3$
Hence
$z_2$ is the G.M of $z_1$ and $z_3$