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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.

Diagram

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  • $\begingroup$ 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. $\endgroup$ – Blue Dec 5 '19 at 9:50

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