Given that \(\omega_1 \) and \(\omega_2\) are two circles which intersects at points \(X\) and \(Y\). Let \(P \) be an arbitrary points on \(\omega_1\). Suppose that the lines \(PX\) and \(PY\) meet \(\omega_2\) again at points \(A\) and \(B\), respectively.

Prove that the circumcircles of all triangles \(PAB\) have the same radius.

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## Comments

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TopNewestProblem 1

We can see that as we move the point \(P\) on the circumference of the circle\([\)excluding \(X\) and \(Y],\)the \(\angle XPY=\angle XP_1Y\) remains constant.So this shows that \(AB=A_1B_1.\)Now we use extended sin rule to complete the problem.

Let the circum-radius of \(\triangle PAB\) be \(R\) and \(\triangle P_1A_1B_1\) be \(R_1.\)

In \(\triangle PAB, \frac{AB}{sin\angle P}=2R\) and in \(\triangle P_1A_1B_1,\frac{A_1B_1}{sin\angle P_1}=\frac{AB}{sin\angle P}=2R_1.\) Therefore \(2R=2R_1\Rightarrow\boxed {R=R_1}.\)Hence Proved.

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Very good solution. Very much clear also. I understood it now.

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Thanks...My fav. is geometry.I see that u like olympiad problems.Can u give some links of ur problems?did u get selected in INMO??

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I have that books in PDF.

Actually I am attending KVS INMO camp for INMO 2017. I have to give INMO directly.

By the way, from which book you do geometry problems?

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My email id : harshus99@gmail.com

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I am currently out of Delhi. I will send you after 5 Nov.

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@Ayush Rai and @Harsh Shrivastava

I have sent IMO books and KVS RMO 2016 paper to your e-mails.

Check your mail.

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Thanks a lot bro.

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@Harsh Shrivastava, @Ayush Rai,

I have 36 Olympiad books related to IMO. Do you want all of them?

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Yeah I also want em all.

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yup.sure.send it to my email.

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Ok no problem.

Can you tell me the author of co-exter book?

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This is there in IF Sharygin, plane geometry.

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A problem copied from IF Sharygin Plane geometry. This is bad.

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Why will I copy? This is RMO Delhi's problem

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What I am saying and what are you understanding.

I am blaming RMO people that they have took this problem from a popular book which usually does not happen.

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This is disappointing :(

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rmo dehli right

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yeah.Are you appearing for this year's rmo?

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