Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.5 AIME, harder Find the answer

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)

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

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.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.