86. The opposite sides of a quadrilateral inscribed in a circle intersect at points $P$ and $Q$. Find the length of the segment $|P Q|$, if the tangents to the circle drawn from $P$ and $Q$ are equal to $a$ and $b$.
86. Let (Fig. 19) $ABCD$ be a cyclic quadrilateral. Describe a circle around $\triangle ADP$. Denote by $M$ the point of intersection of this circle with the line $PQ$. We have $\widehat{DMQ}=\widehat{DAP}=\widehat{BCD}$. Therefore, the quadrilateral $CDMQ$ is cyclic. Since, by the condition, the tangents drawn from $P$ and $Q$ to the original circle are equal to $a$ and $b$, we will have

$$
\begin{aligned}
& |QM| \cdot |QP_{\mid}=QD| \cdot |QA|=b^{2} \\
& |PM| \cdot |PQ|=|PD| \cdot |PC|=a^{2}
\end{aligned}
$$

Adding these equalities, we get

$$
|PQ|^{2}=a^{2}+b^{2}, \quad|PQ|=\sqrt{a^{2}+b^{2}}.
$$