# Circles

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. Note by Sayantan Saha
4 years, 1 month ago

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Sort by: Problem 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.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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??

- 4 years, 1 month ago

I have many interesting books regarding IMO. I post problems from there only.

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?

- 4 years, 1 month ago

Sometimes co-exter or pre-college maths.Can u give the pdf of the books to this email : raiayush234@gmail.com

- 4 years, 1 month ago

and also give the INMO paper to this email.Did u write RMO this year

- 4 years, 1 month ago

Yes I appeared in RMO this year from KVS region.

- 4 years, 1 month ago

How many did u solve?

- 4 years, 1 month ago

I solved 4 and half.

- 4 years, 1 month ago

Awesome! Send me the question paper also then

- 4 years, 1 month ago

You will get it.

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

- 4 years, 1 month ago

ok.no probs. Can u give ur email address

- 4 years, 1 month ago

priyanshu_2feb@rediffmail.com

- 4 years, 1 month ago

Please share these resources with me also.

My email id : harshus99@gmail.com

- 4 years, 1 month ago

rmo dehli right

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

The inequality problem in Delhi region was also copied.I don't remember the source, but I had seen it before RMO on some forum.

This is disappointing :(

- 4 years, 1 month ago

Actually this year Delhi paper was prepared by Delhi coordinator only not by HBCSE. So he copied many things because framing a question requires a lot of time.

- 4 years, 1 month ago

This is there in IF Sharygin, plane geometry.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

yup.sure.send it to my email.

- 4 years, 1 month ago

Ok no problem.

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

- 4 years, 1 month ago

H. S. M. Coxeter and S. L. Greitzer

- 4 years, 1 month ago

Is it Geometry revisited by MAA?

- 4 years, 1 month ago

yup

- 4 years, 1 month ago

That is not so good. It's good for reading only. It has less computational problems.

- 4 years, 1 month ago

yup.i use it to learn new stuff.

- 4 years, 1 month ago

Do you have any good book regarding NUMBER THEORY?

- 4 years, 1 month ago

David M.burton book;s

- 4 years, 1 month ago

can u give ur email ID

- 4 years, 1 month ago

I gave. See above.

- 4 years ago

Yeah I also want em all.

- 4 years ago

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

- 4 years ago

Thanks a lot bro.

- 4 years ago