# Similarity coefficient of two inverted circles

The coefficient of similarity between two circles $$C$$ and $$C’$$ of radii $$r$$ and $$r’$$ is $$\frac{r}{r’}=\frac {k}{k_n},$$ where $$k$$ is the radius of inversion and $$k_n$$ is the square of the length of a tangent to $$C’$$.

Why is this the case? I spent a lot of time trying to show it geometrically, but I didn't succeed. Here is an image of the construction.

• I don't believe the result is correct as stated. What is true is that $$\frac{r}{r'}=\left(\frac{t}{k}\right)^2=\left(\frac{k}{t'}\right)^2$$ where $k$ is the radius of inversion, and $t$ and $t'$ are tangent segments from the center of inversion to $C$ and $C'$, respectively. – Blue Dec 5 '19 at 9:50