13. As shown in Figure 6, a right trapezoid $A O C D$ is placed in the Cartesian coordinate system. It is known that $A D=3, A O=8, O C=5$. If point $P$ is inside the trapezoid, and
$$
\begin{array}{l}
S_{\triangle P A D}=S_{\triangle P O C}, \\
S_{\triangle P A O}=S_{\triangle P C D},
\end{array}
$$

then, the coordinates of point $P$ are
13. $\left(\frac{17}{8}, 3\right)$.
$S_{\text {trapezoid } A O C D}=\frac{(3+5) \times 8}{2}=32$.
As shown in Figure 14, draw $P E$ $\perp y$-axis at point $E$.

Since $S_{\triangle P R D}=S_{\triangle P Q C}$, we have $3 A E=5 O E$, that is,
$$
3(8-O E)=5 O E \text {. }
$$

Solving this, we get $O E=3$.
Thus, $S_{\triangle P A D}=S_{\triangle P O C}=7.5$,
$$
\begin{array}{l}
S_{\triangle P A O}=S_{\triangle P C D} \\
=(32-2 \times 7.5) \div 2=8.5 .
\end{array}
$$

Therefore, $\frac{1}{2} \times 8 P E=8.5$, which means $P E=\frac{17}{8}$.