10. $P$ is a point inside $\triangle ABC$, and $D, E, F$ are the feet of the perpendiculars from $P$ to $BC, CA, AB$ respectively. Find all points $P$ that minimize $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$. (22nd IMO Problem)
10. Let the three sides of $\triangle A B C$ be $A B=c, B C=a, C A=b$, the area be $S, P D=x, P E=y$, $P F=z$, then $a x+b y+c z=2 S$. By the Cauchy-Schwarz inequality, we have
$$\begin{array}{l}
\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\right)(a x+b y+c z) \geqslant(a+b+c)^{2} \Rightarrow \\
\frac{a}{x}+\frac{b}{y}+\frac{c}{z} \geqslant \frac{(a+b+c)^{2}}{2 S}
\end{array}$$

The equality holds if and only if $x=y=z$, i.e., when $P$ is the incenter of $\triangle A B C$, the expression reaches its minimum.