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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#Geometry #RMO #Proofs #Triangles #Circles

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Comments

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I got this! I hope it is right.

Ranjana Kasangeri - 6 years, 2 months ago

How?

Aneesh Kundu - 6 years, 2 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!

Ranjana Kasangeri - 6 years, 2 months ago

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

Satvik Golechha - 6 years, 2 months ago

I solved 4

Aneesh Kundu - 6 years, 2 months ago

Dunno whether they are right or wrong

Aneesh Kundu - 6 years, 2 months ago

can someone give me the solution to this problem? plzz

Nihar Mahajan - 6 years, 2 months ago

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

Satvik Golechha - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$