Problem 4! IMO 2015

Triangle $$ABC$$ has circumcircle $$\Omega$$ and circumcenter $$O$$. A circle $$\Gamma$$ with center $$A$$ intersects the segment $$BC$$ at points $$D$$ and $$E$$, such that $$B$$, $$D$$, $$E$$, and $$C$$ are all different and lie on line $$B$$ in this order. Let $$F$$ and $$G$$ be the points of intersection of $$\Gamma$$ and $$\Omega$$, such that $$A$$, $$F$$, $$B$$, $$C$$, and $$G$$ lie on $$\Omega$$ in this order. Let $$K$$ be the second point of intersection of the circumcircle of triangle $$BDF$$ and the segment $$AB$$. Let $$L$$ be the second point of intersection of the circumcircle of triangle $$CGE$$ and the segment $$AC$$.

Suppose that the lines $$FK$$ and $$GL$$ are different and intersect at the point $$X$$. Prove that $$X$$ lies on the line $$AO$$.

This is part of the set IMO 2015

Note by Sualeh Asif
2 years, 11 months ago

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This is a problem which looks initially scary with all the circles, but is just a chasing down of angles. Work backwards from the result, and see what we need. E.g. If X lies on AO, what does that tell us about X?

Staff - 2 years, 11 months ago