4. On one of the sides of the right triangle $A B C$, whose legs are equal to 1, a point $P$ is chosen. Find the maximum value of $P A \cdot P B \cdot P C$.

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
4. On one of the sides of the right-angled triangle $ABC$, with legs equal to 1, a point $P$ is chosen. Find the maximum value of $PA \cdot PB \cdot PC$.

$\square \mathrm{B}$ in the Cartesian coordinate system $A(1,0) ; B(0,1) ; C(0,0)$. If $P(x, 0)$ is on side $AC$, then

$$
PA \cdot PB \cdot PC = x(1-x) \sqrt{1+x^{2}} \leq \frac{1}{4} \sqrt{2}
$$

but this estimate is not achieved for $x \in [0,1]$. A similar situation occurs if $P$ is on side $BC$. Let $P(x, 1-x)$ be on the hypotenuse $AB$. Then

$$
PA \cdot PB \cdot PC = 2x(1-x) \sqrt{x^{2}+(1-x)^{2}} = 2t \sqrt{1-2t}
$$

where $t = x(1-x) \in [0,1/4]$. The function $2t \sqrt{1-2t}$ is increasing on the interval $[0,1/4]$, so $\max PA \cdot PB \cdot PC = \sqrt{2}/4$ when $t = 1/4, x = 1/2$. The point $P$ is the midpoint of the hypotenuse in this case.