# 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
3 years, 4 months ago

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Quite similar to one which came in Rajasthan paper, though more difficult.

- 3 years, 4 months ago

can someone give me the solution to this problem? plzz

- 3 years, 4 months ago

How many are you getting right? (Outta 6, right?)

- 3 years, 4 months ago

Dunno whether they are right or wrong

- 3 years, 4 months ago

I solved 4

- 3 years, 4 months ago

I got this! I hope it is right.

- 3 years, 4 months ago

How?

- 3 years, 4 months ago

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!

- 3 years, 4 months ago