13. A circle is inscribed in $\triangle A B C$ with sides $A B=4, B C=6$, and $A C=8$. If $P$ and $Q$ are the respective points of tangency of $\overline{A B}$ and $\overline{A C}$ with the circle, determine the length of chord $P Q$.
13. $\frac{3 \sqrt{10}}{4}$
Applying the Law of Cosines to $\triangle A B C$, we get
$$
\cos A=\frac{4^{2}+8^{2}-6^{2}}{2 \cdot 4 \cdot 8}=\frac{11}{16} .
$$

Let $A P=A Q=x, P B=y$, and $Q C=z$. Then we have the following system of equations:
$$
\left\{\begin{array}{l}
x+y=4 \\
x+z=8 \\
y+z=6 .
\end{array}\right.
$$

Adding the equations, we get $2 x+2 y+2 z=18$ or $x+y+z=9$, so that $x=(x+y+z)-(y+z)=9-6=3, y=1$, and $z=5$. Finally, applying the Law of Cosines to $\triangle A P Q$, we have
$$
P Q^{2}=A P^{2}+A Q^{2}-2 A P \cdot A Q \cos A=3^{2}+3^{2}-2 \cdot 3 \cdot 3 \cdot \frac{11}{16}=\frac{45}{8}
$$