Let z1, z2, z3 be the vertices of A,B,C of a triangle with the origin(O) the circumcenter. A transformation takes ABC to its circumcircle fixing the center and with the following property:-

A point on AB be a.z1+(1-a)z2. Then after transformation it will be z1^a.z2^(1-a)

Similar for BC and CA.

This takes ABC to its circumcircle.

By homothety with respect to O this transformation is defined for all other points in the plane also.

Can anyone tell me what a line not passing through O changes to by this transformation??