Competition problem — Geometry — Maths Olympiad Prep

Library / /1 of 94

Geometry Difficulty 4.5 AIME Prove it Hong Kong

The incircle of a triangle $ABC$ touches the three sides $BC$ , $CA$ and $AB$ at $D$ , $E$ and $F$ respectively. The line segments $BE$ and $CF$ intersect the incircle at $Y$ and $Z$ respectively. Suppose the lines $FE$ and $BC$ intersect externally at $X$ . Prove that the three points $X$ , $Y$ and $Z$ are collinear.

Solution

Let $X' = EF \cap YZ$ . Apply Pascal's theorem to the points $EEFFZY$ on the incircle. Then $C = EE \cap FZ$ , $X' = EF \cap ZY$ and $B = FF \cap YE$ are collinear. Hence, $X' = X$ , and $X, Y, Z$ are collinear.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.