Let $z_1,z_2,\mathrm{.......},z_n$  be in G.P. with first term as unity such that $z_1+z_2+\mathrm{.......}+z_n=0.$  Now if $z_1,z_2,\mathrm{.....}{z}_n$  represents vertices of a polygon, then distance between the incentre and circumcentre of the polygon is:

Let $r$  be the common ratio of the G.P.

$z_1,z_2,\mathrm{......},z_n$ with $z_1=1.$

$z_1+z_2+\mathrm{.....}+z_n=0$ $\Rightarrow \frac{\left(r^n-1\right)}{r-1}=0$ $\left(r\ne 1\right)$

$\Rightarrow$           $r^n=1=\left|r\right|=1$

$\Rightarrow$           $\left|z_k\right|=\left|r^k\right|={\left|r\right|}^k=1$  

$\Rightarrow$           The polygon $z_1,z_2,\mathrm{......},z_n$ is regular

$\Rightarrow$           in centre and circumcentre are same

$\Rightarrow$           distance between incentre and circumcentre is zero