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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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.

Log in to reply

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

Log in to reply

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??

Log in to reply

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?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

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

Log in to reply

Log in to reply

Log in to reply

My email id : harshus99@gmail.com

Log in to reply

rmo dehli right

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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.

Log in to reply

This is disappointing :(

Log in to reply

Log in to reply

This is there in IF Sharygin, plane geometry.

Log in to reply

@Harsh Shrivastava, @Ayush Rai,

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

Log in to reply

yup.sure.send it to my email.

Log in to reply

Ok no problem.

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

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Yeah I also want em all.

Log in to reply

@Ayush Rai and @Harsh Shrivastava

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

Check your mail.

Log in to reply

Thanks a lot bro.

Log in to reply