[ Touching Circles ]

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Find the ratio of the radii of two circles touching each other, if each of them touches the sides of an angle equal to $\alpha$.

#
Drop a perpendicular from the center of the smaller circle to the radius of the larger circle, drawn to the point of tangency with one of the sides of the angle.

## Solution

Let $r$ and $R$ be the radii of the circles ($R > r$). Draw the radius of the larger circle to the point of tangency with one of the sides of the angle and drop a perpendicular from the center of the smaller circle to this radius. We obtain a right triangle with the hypotenuse $R + r$, one leg $R - r$, and the acute angle opposite this leg equal to $\frac{\alpha}{2}$.

Thus,

$$
\frac{R-r}{R+r}=\sin \frac{\alpha}{2}
$$

From this, we find that

$$
\frac{r}{R}=\frac{1-\sin \frac{\alpha}{2}}{1+\sin \frac{\alpha}{2}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_93426a107eaf3f76289cg-37.jpg?height=437&width=1018&top_left_y=1885&top_left_x=520)

Answer

$\frac{1-\sin \frac{\alpha}{2}}{1+\sin \frac{\alpha}{2}}$.