[ $[$ Tangent Circles $]$

In the sector $A O B$ with radius $R$ and angle $90^{\circ}$, a circle is inscribed, touching segments $O A, O B$ and arc $A B$. Find the radius of the circle.

#
The line of centers of touching circles passes through the point of tangency.

## Solution

Let $x$ be the radius of the desired circle, $O_{1}$ its center, and $M$ the point of tangency of the smaller circle with radius $OB$. In triangle $O O_{1} M$, it is known that

$$
M O_{1}=x, O O_{1}=R-x, \angle O O_{1} M=45^{\circ}
$$

Therefore, $R-x=x \sqrt{2}$. Hence,

$$
x=\frac{R}{\sqrt{2}+1}=R(\sqrt{2}-1)
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-18.jpg?height=428&width=420&top_left_y=2470&top_left_x=822)

$$
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## and other.]

Classes: 8,9 $\qquad$ $\qquad$ $\qquad$