Consider two given circles of radii $r_1$ and $r_2$ with centres $C_1$ and $C_2$. A point $P$ is such that $\frac{r_1}{r_2} = \frac{PC_1}{PC_2}$. I wanted to know how the locus of $P$ would look like.
There would be such a point on the line segment $C_1C_2$ and if we drew a random line through that $P$ then there would be another point(say $P'$) such that $P'P$ would bisect $\angle C_1P'C_2$. This point would also be included in the locus. This is not the only way to find out the locus though, there is even the homothetic centre and maybe many more such points.
All in all, I want to know the shape of the locus and if possible, the method I could use to find it out without using a computer.