2019 IMO SL #G1

 Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

We have\[ \measuredangle TGF = \measuredangle TGC = \measuredangle GEC = \measuredangle GEA = \measuredangle GFA \]and similarly $\measuredangle TFG = \measuredangle FGA$. Thus $TAGF$ is an isosceles trapezoid, as needed. 

This has got to be the easiest IMO Shortlist problem I've ever seen.


  1. Anonymous7/06/2022

    i know, right? this one was way too easy lol


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