APMO

Let $h_a$ and $h_b$ be the altitudes from $A$ and $B$, respectively. Then $$\begin{array}{rl}AB\cdot h\cdot AC\cdot h_b\cdot BC\cdot h_a& =8\cdot (\text{area of }△ABC{)}^3\\ & =(AB\cdot h{)}^3,\end{array}$$ which is a constant. So the product $h\cdot h_a\cdot h_b$ attains its maximum when the product $AC\cdot BC$ attains its minimum.

Since $$\begin{array}{rl}(\sin C)\cdot AC\cdot BC& =BC\cdot h_a\\ & =2\cdot \text{area of }△ABC,\end{array}$$ which is a constant, $AC\cdot BC$ attains its minimum when $\sin C$ reaches its maximum. There are two cases:

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

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