Three. (25 points) The parabola $y=a x^{2}+b x+c(a>$ $0)$ passes through points $A(-3 \sqrt{3}, 0)$ and $B(\sqrt{3}, 0)$, and intersects the $y$-axis at point $C$, with $\angle A C B \geqslant 90^{\circ}$. Let the vertex of the parabola be $D$, and in $\triangle B C D$, the altitude from $C$ to $D$ is $h$. Find the range of values for $a$ and $h$.

---

The parabola $y=a x^{2}+b x+c(a>0)$ passes through points $A(-3 \sqrt{3}, 0)$ and $B(\sqrt{3}, 0)$, and intersects the $y$-axis at point $C$, with $\angle A C B \geqslant 90^{\circ}$. Let the vertex of the parabola be $D$, and in $\triangle B C D$, the altitude from $C$ to $D$ is $h$. Find the range of values for $a$ and $h$.
Three, because $A(-3 \sqrt{3}, 0)$ and $B(\sqrt{3}, 0)$ are on $y=a x^{2}+b x+c$, then
$$
y=a(x+3 \sqrt{3})(x-\sqrt{3})=a\left(x^{2}+2 \sqrt{3} x-9\right) \text {. }
$$

Thus, $c=-9 a$.
When $\angle A C B=90^{\circ}$, by the projection theorem we get
$$
O C^{2}=O A \cdot O B=3 \sqrt{3} \times \sqrt{3}=9 \Rightarrow O C=3 \text {. }
$$

Since $\angle A C B \geqslant 90^{\circ}$, it follows that $O C \leqslant 3$.
Therefore, $9 a \leqslant 3, a \leqslant \frac{1}{3}$.
Also, $a>0$, so, $0<a \leqslant \frac{1}{3}$.

As shown in Figure 10, draw $D E \perp O C$, extend $D C$ to intersect the $x$-axis at point $H$, and draw $B F \perp C H$ at point $F$.

Since $D$ is the vertex of the parabola, we have
$$
D(-\sqrt{3},-12 a) \text {. }
$$

Thus, $O E=12 a$.
Also, $O C=9 a$, so $C E=3 a, D E=\sqrt{3}$.
It is easy to prove that $\triangle H C O \backsim \triangle D C E$, hence
$$
\frac{H O}{D E}=\frac{C O}{C E}=\frac{9 a}{3 a}=3 .
$$

Therefore, $O H=3 D E=3 \sqrt{3}, B H=O H-O B=2 \sqrt{3}$.
Since $O C \leqslant 3$, then $\tan \angle O H C=\frac{O C}{O H} \leqslant \frac{3}{3 \sqrt{3}}=\frac{\sqrt{3}}{3}$.
Thus, $0^{\circ}<\angle O H C \leqslant 30^{\circ}$.
Then, $h=B F=B H \sin \angle B H F \leqslant B H \sin 30^{\circ}=\sqrt{3}$.
Therefore, $0<h \leqslant \sqrt{3}$.
Hence, $0<a \leqslant \frac{1}{3}, 0<h \leqslant \sqrt{3}$.
(Zou Shouzhang, Anhui Province Nanfei County Experimental Middle School, 241300)