Let z1,z2,.......,zn  be in G.P. with first term as unity such that z1+z2+.......+zn=0.  Now if z1,z2,.....zn  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. z1,z2,......,zn with z1=1. z1+z2+.....+zn=0 ⇒(rn−1)r−1=0 (r≠1) ⇒           rn=1=|r|=1 ⇒           |zk|=|rk|=|r|k=1  ⇒           The polygon z1,z2,......,zn is regular⇒           in centre and circumcentre are same⇒           distance between incentre and circumcentre is zero