In a circle with center $O$ and radius $r$, let $A B$ be a chord smaller than the diameter. The radius of the circle inscribed in the smaller sector determined by the radii $A O$ and $B O$ is $\varrho$. Express $A B$ in terms of $r$ and $\varrho$.
A circle with radius $\varrho$ has its center denoted by $O_{1}$, and it touches the segment $O B$ at point $D$. Let the midpoint of segment $A B$ be $F$. The triangle $O F B$ is similar to the triangle $O O_{1} D$, because both are right-angled, and their $\angle F O D$ is common. From the similarity,

$$
\frac{A B}{2}: \varrho=r:(r-\varrho), \text { hence } A B=\frac{2 r \varrho}{r-\varrho} \text {. }
$$

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