Track / Stage 5 / 38 of 400 #638 of 2000

Problem 638

AIME late

Geometry Difficulty 5.1 Prove it

Example 2. As shown in Figure 2, from the vertex $A$ of $\triangle ABC$ , draw any line intersecting the excircle near $BC$ at $P$ and $Q$ . Prove: $AP + AQ \geqslant AB + BC + CA$

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

$\begin{array}{l} \text { Prove } A P+A Q \\ \geqslant 2 \sqrt{A P \cdot A Q} \\ = 2 A D . \\ \text { But } 2 A D=A D+A E \\ = A B+B D \\ + A C+C B \\ = A B+B F \\ +A C+C F \\ = A B+B C+A C \\ \therefore \quad A P+A Q \geqslant A B+B C+C A .\end{array}$

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