3. Consider all bases $AB$ fixed, and point $C$ drawn such that the height of $\triangle ABC$ is a constant $h$. When is the product of its heights maximized?

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3. Since $h_{\mathrm{a}} \cdot a=h_{\mathrm{b}} \cdot b=h \cdot c$ is a constant, we have
$$
k_{\mathrm{a}} h_{\mathrm{b}} h_{h}=\frac{h^{3} \cdot c^{2}}{a b},
$$

it is only necessary to minimize $a b$.
When $a=b$, $\triangle A B C$ is an isosceles triangle. Let this point be $C$ and $C_{0}$, and the tangent line of the circumcircle $\odot O$ at point $C_{0}$ be $l$, then $l \parallel A B$ and the distance from $l$ to $A B$ is $/$.
- The vertex $C$ of $\triangle A B C$ must lie on $l$, because: $l$ is the tangent line, $C$ is outside $\odot O$, $\angle A C B \cdot \frac{c h}{\sin \angle A C_{0} B} \\
& =C_{0} A \times C_{0} B .
\end{aligned}
$$

That is, when $C$ coincides with $C_{0}$, $a b$ is minimized.