Asian Pacific Mathematics Olympiad

Let $h_{a}$ and $h_{b}$ be the altitudes from $A$ and $B$, respectively. Then
$$\begin{aligned} A B \cdot h \cdot A C \cdot h_{b} \cdot B C \cdot h_{a} &= 8 \cdot (\text{area of } \triangle A B C)^3 \\ &= (A B \cdot h)^3, \end{aligned}$$
which is a constant. So the product $h \cdot h_{a} \cdot h_{b}$ attains its maximum when the product $A C \cdot B C$ attains its minimum.

Since
$$\begin{aligned} (\sin C) \cdot A C \cdot B C &= B C \cdot h_{a} \\ &= 2 \cdot \text{area of } \triangle A B C, \end{aligned}$$
which is a constant, $A C \cdot B C$ attains its minimum when $\sin C$ reaches its maximum. There are two cases:

a. $h \leq A B / 2$. Then there exists a triangle $A B C$ which has a right angle at $C$, and for precisely such a triangle $\sin C$ attains its maximum, namely $1$.

b. $h > A B / 2$. In this case the angle at $C$ is acute and assumes its maximum when the triangle is isosceles.