16. As shown in Figure 6, given point $P$ is moving on the arc $\overparen{A B}$ (excluding endpoints) of a sector $O A B$ with a radius of 6 and a central angle of $90^{\circ}$, $P H \perp O A$, with the foot of the perpendicular at $H$, and the centroid of $\triangle O P H$ is $G$.
(1) When point $P$ is moving on $\overparen{A B}$, are there any line segments among $G O$, $G P$, and $G H$ whose lengths remain constant? If so, please indicate which one and find its corresponding length;
(2) Let $P H=x$, $G P=y$, find the function expression of $y$ in terms of $x$, and write the range of the variable $x$;
(3) If $\triangle P G H$ is an isosceles triangle, try to find the length of line segment $P H$.
16. (1) Online destroy $G O$, $C$, $G H$ Ning, there is a line segment, which is $\mathrm{UH}$, whose length remains unchanged.
Extend $HQ$ to intersect $O$ at $E$, and extend $PG$ to intersect $AO$ at $D$.
$\because G$ is the centroid of $\triangle OPH$; and $\angle PHO=90^{\circ}$,
$$
\therefore GH=\frac{2}{3} HE=\frac{2}{3} \times \frac{1}{2} \times GP=\frac{1}{3} \times 6=2 \text{. }
$$
(2) In the right triangle $\triangle OPH$, we have
$$
\begin{array}{l}
OH=\sqrt{OP^{2}-PH^{2}}=\sqrt{36-x^{2}}, \\
DH=\frac{1}{2} OH=\frac{1}{2} \sqrt{36-x^{2}}.
\end{array}
$$

In the right triangle $\triangle DPH$,
$$
\begin{aligned}
\because DP & =\sqrt{DH^{2}+PH^{2}} \\
& =\sqrt{\frac{1}{4}\left(36-x^{2}\right)+x^{2}} \\
& =\frac{1}{2} \sqrt{36+3 x^{2}}, \\
\therefore y= & GP=\frac{2}{3} DP=\frac{1}{3} \sqrt{36+3 x^{2}}(0<x<6).
\end{aligned}
$$
(3) $\triangle PGH$ is an isosceles triangle, with three possible cases:
(1) $GP=PH$, i.e., $\frac{1}{3} \sqrt{36+3 x^{2}}=x$, solving gives $x=\sqrt{6}$, which is a valid root and meets the problem's requirements;
(2) $GP=GH$, i.e., $\frac{1}{3} \sqrt{36+3 x^{2}}=2$, solving gives $x=0$, which is a valid root but does not meet the problem's requirements;
(3) $PH=GH$, i.e., $PH=2$.
In summary, if $\triangle PGH$ is an isosceles triangle, then the length of segment $PH$ is $\sqrt{6}$ or 2.