7ᵗʰ Iranian Geometry Olympiad (Intermediate) P3

For discussing Olympiad level Geometry Problems
IftakharTausifFarhan
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Joined:Sat Dec 12, 2020 3:33 pm
7ᵗʰ Iranian Geometry Olympiad (Intermediate) P3

Unread post by IftakharTausifFarhan » Sat Dec 12, 2020 4:13 pm

In acute-angled triangle $ABC (AC \gt AB)$, point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid $(CB \parallel HJ)$. Prove that $JB$ and $EM$ meet on $\omega$.

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