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RMO 2014 Delhi Region Q.5

Let \(ABC\) be an acute-angeled triangle and let \(H\) be its orthocenter. For any point \(P\) on the circumcircle of triangle \(ABC\), let \(Q\) be the point of the intersection of the line \(BH\) with the line \(AP\) . Show that there is a unique point \(X\) on the circumcircle of \(ABC\) such that for every point \(P\not=A,B\) the circumcircle of \(HQP\) pass through \(X\).


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Note by Aneesh Kundu
2 years, 5 months ago

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Quite similar to one which came in Rajasthan paper, though more difficult. Satvik Golechha · 2 years, 5 months ago

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can someone give me the solution to this problem? plzz Nihar Mahajan · 2 years, 5 months ago

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How many are you getting right? (Outta 6, right?) Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha Dunno whether they are right or wrong Aneesh Kundu · 2 years, 5 months ago

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@Satvik Golechha I solved 4 Aneesh Kundu · 2 years, 5 months ago

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I got this! I hope it is right. Ranjana Kasangeri · 2 years, 5 months ago

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@Ranjana Kasangeri How? Aneesh Kundu · 2 years, 5 months ago

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@Aneesh Kundu Take any other point say, \(Z\) and let \(AZ\) intersect \(BH\) at \(Y\). And consider circle \(HYZ\) meeting circle \(ABC\) at \(X.\) Now,prove \(HQPX\) is a cyclic quad! Ranjana Kasangeri · 2 years, 5 months ago

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