Track / Stage 5 / 200 of 400 #800 of 2000

Problem 800

AIME late

Geometry Difficulty 5.5 Prove it

Example 3. $P$ is any point inside $\triangle ABC$ , $AP$ , $BP$ , $CP$ intersect the opposite sides at $P_{1}$ , $P_{2}$ , $P_{3}$ respectively. Prove that $\frac{AP}{PP_{1}}$ , $\frac{BP}{PP_{2}}$ , $\frac{CP}{PP_{3}}$ are such that at least one is not less than 2, and at least one is not greater than 2. (3rd IMO problem)

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Proof As shown in the figure, let $G$ be the centroid, $G A^{\prime} \| B C$ , $G B^{\prime} \| A C$ , $G C^{\prime} \| A B$ , which divides $\triangle A B C$ into three trapezoids. When $P$ lies on $A A^{\prime} G C^{\prime}$ (including the interior), $\frac{A P}{P P_{1}} \leqslant \frac{A M}{M P_{I}}=\frac{A G}{G K}=2$ , and $\frac{C P}{P P_{8}} \geqslant \frac{C N}{N P_{3}}=\frac{C G}{G L}=2$ . When $P$ lies on the other two trapezoids, the same reasoning applies.

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