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China Hong Kong Mathematical Olympiad

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.

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