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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, 6 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?

Calvin Lin Staff - 2 years, 6 months ago

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