Iranian Geometry Olympiad 2020 (Advanced) P4

For discussing Olympiad level Geometry Problems
IftakharTausifFarhan
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Joined:Sat Dec 12, 2020 3:33 pm
Iranian Geometry Olympiad 2020 (Advanced) P4

Unread post by IftakharTausifFarhan » Sat Dec 12, 2020 4:35 pm

Convex circumscribed quadrilateral $ABCD$ with incenter $I$ is given such that its incircle is tangent to $AD$, $DC$, $CB$, and $BA$ at $K$, $L$, $M$, and $N$. Lines $AD$ and $BC$ meet at $E$ and lines $AB$ and $CD$ meet at $F$. Let $KM$ intersects $AB$ and $CD$ at $X$ and $Y$ , respectively. Let $LN$ intersects $AD$ and $BC$ at $Z$ and $T$, respectively. Prove that the circumcircle of triangle $XF Y$ and the circle with diameter $EI$ are tangent if and only if the circumcircle of triangle $TEZ$ and the circle with diameter $FI$ are tangent.

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