BdMO 2013 Higher Secondary Problem 9

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FahimFerdous
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BdMO 2013 Higher Secondary Problem 9

Unread post by FahimFerdous » Mon Feb 11, 2013 3:53 pm

Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AD, BE, CF$ is a diametre. Extended $AB$ and $DC$ meet at $Z$, $CD$ and $FE$ at $X$, $EF$ and $BA$ at $Y$. $AC$ and $BF$ meets at $P$, $CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R$. If $O$ is the point of intersection of $YQ$ and $ZR$, find $\angle XOP$.
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FahimFerdous
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Re: BdMO 2013 Higher Secondary Problem 9

Unread post by FahimFerdous » Mon Feb 11, 2013 4:01 pm

Let $AD \cap EF=T$, $BE \cap CD=S$, $AB \cap CF=K$, $AE \cap BD=L$, $CR \cap FQ=M$. Then use Pascal's theorem and Desaurge's theorem to prove that $P, Q, T; P, R, S; R, Q, K; X, O, M; L, M, P; X, L, P$ are collinear (I'm leaving this tedious job to you guys, I can't write it). And it implies that $X, O, P$ are collinear.

I still can't believe I missed this and problem 8. :(
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sourav das
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Re: BdMO 2013 Higher Secondary Problem 9

Unread post by sourav das » Mon Feb 11, 2013 5:27 pm

You spin my head right round right round,
When you go down, when you go down down......
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*Mahi*
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Re: BdMO 2013 Higher Secondary Problem 9

Unread post by *Mahi* » Mon Feb 11, 2013 9:52 pm

Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

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