91. In quadrilateral $A B C D$, given are $|A B|=a$, $|A D|=b$; sides $B C, C D$ and $A D$ touch a certain circle, the center of which is at the midpoint of $A B$. Find the side $|B C|$.
91. $D O$ and $C O$ are the bisectors of angles $A D C$ and $D C B$. Let $\alpha, \beta$ and $\gamma$ be the measures of the corresponding angles (Fig. 22). But $\alpha+2 \beta+2 \gamma+\alpha=2 \pi$, hence, $\alpha+\beta+\gamma=\pi$; from this it follows that $\widehat{D O A}=\gamma, \widehat{C O B}=\beta$ and $\triangle A O D$ is similar to $\triangle C O B$, from which

$$
\begin{aligned}
& \frac{|A D|}{|A O|}=\frac{|O B|}{|C B|}, \quad|A D| \cdot|C B|=|A O| \cdot|O B|=\frac{|A B|^{2}}{4} \\
& \text { Answer: } \frac{a^{2}}{4 b}
\end{aligned}
$$