[ Touching Circles ]

[

The central angle of the sector is $60^{\circ}$, and the radius is $R$. Find the radius of the circle inscribed in this sector.

#
The line connecting the centers of two tangent circles passes through their point of tangency.

## Solution

Let the desired radius be $x$. Since the line connecting the centers of two tangent circles passes through their point of tangency, and the leg opposite the $30^{\circ}$ angle is half the hypotenuse, we have $R - x = 2x$. From this, we find that $x = \frac{R}{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-14.jpg?height=446&width=512&top_left_y=0&top_left_x=773)

## Answer

$\frac{R}{3}$.