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Thread: IMO Problem II

  1. #1
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    IMO Problem II

    This problem comes from this year's International Mathematical Olympiad , comparing with another geometry problem on the next day (problem 4) , this seems a little harder . Enjoy !

    Let $\displaystyle I $ be the incentre of triangle $\displaystyle ABC$ and let $\displaystyle \Gamma $ be its circumcircle. Let the line $\displaystyle AI $ intersect $\displaystyle \Gamma $ again at $\displaystyle D $ . Let $\displaystyle E $ be a point on the arc $\displaystyle BDC $ and $\displaystyle F $ a point on the side $\displaystyle BC $ such that

    $\displaystyle \angle BAF = \angle CAE < \frac{1}{2} \angle BAC $ .

    Finally , let $\displaystyle G $ be the midpoint of the segment $\displaystyle IP $ . Prove that the lines $\displaystyle DG $ and $\displaystyle EI $ intersects on $\displaystyle \Gamma $ .
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    Senior Member BAdhi's Avatar
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    sir, sorry for the late,late reply. now only i saw this thread

    Quote Originally Posted by simplependulum View Post
    Finally , let $\displaystyle G $ be the midpoint of the segment $\displaystyle IP $ .
    could you tell me what is $\displaystyle P$ here?
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    Quote Originally Posted by BAdhi View Post
    sir, sorry for the late,late reply. now only i saw this thread



    could you tell me what is $\displaystyle P$ here?
    In order that DG and EI intersect on the circumcircle,
    G ought to be the midpoint of IF,
    so I guess that's a typo.
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  4. #4
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    Yes , it is a typo . It should be $\displaystyle IF $ not $\displaystyle IP $ .
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