Iranian Geometry Olympiad 2020 (Advanced) P5

For discussing Olympiad level Geometry Problems
IftakharTausifFarhan
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Iranian Geometry Olympiad 2020 (Advanced) P5

Unread post by IftakharTausifFarhan » Sat Dec 12, 2020 4:45 pm

Consider an acute-angled triangle $ABC (AC \gt AB)$ with its orthocenter $H$ and circumcircle $\Gamma$. Points $M$ and $P$ are the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $BC$ so that $NX$ is tangent to $\Gamma$. Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP = \angle HNM$ ($B$ and $J$ lie on the same side of $AH$) and circle $\omega_{1}$, passing through $K$, $H$, and $J$, and circle $\omega_{2}$, passing through $K$, $M$, and $N$, are externally tangent to each other. Prove that the common external tangents of $\omega_{1}$ and $\omega_{2}$ meet on the line $NH$.

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