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# 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
7 months, 3 weeks ago

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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. · 7 months, 1 week ago

Very good solution. Very much clear also. I understood it now. · 7 months 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?? · 7 months 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? · 7 months ago

Please share these resources with me also.

My email id : harshus99@gmail.com · 7 months ago

and also give the INMO paper to this email.Did u write RMO this year · 7 months ago

Yes I appeared in RMO this year from KVS region. · 7 months ago

How many did u solve? · 7 months ago

I solved 4 and half. · 7 months ago

Awesome! Send me the question paper also then · 7 months ago

You will get it.

I am currently out of Delhi. I will send you after 5 Nov. · 7 months ago

ok.no probs. Can u give ur email address · 7 months ago

priyanshu_2feb@rediffmail.com · 7 months ago

Sometimes co-exter or pre-college maths.Can u give the pdf of the books to this email : raiayush234@gmail.com · 7 months ago

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

Check your mail. · 6 months, 3 weeks ago

Thanks a lot bro. · 6 months, 3 weeks ago

I have 36 Olympiad books related to IMO. Do you want all of them? · 7 months ago

Yeah I also want em all. · 7 months ago

yup.sure.send it to my email. · 7 months ago

Ok no problem.

Can you tell me the author of co-exter book? · 7 months ago

H. S. M. Coxeter and S. L. Greitzer · 7 months ago

Is it Geometry revisited by MAA? · 7 months ago

yup · 7 months ago

That is not so good. It's good for reading only. It has less computational problems. · 7 months ago

yup.i use it to learn new stuff. · 7 months ago

Do you have any good book regarding NUMBER THEORY? · 7 months ago

can u give ur email ID · 7 months ago

I gave. See above. · 7 months ago

David M.burton book;s · 7 months ago

This is there in IF Sharygin, plane geometry. · 7 months, 2 weeks ago

A problem copied from IF Sharygin Plane geometry. This is bad. · 7 months, 2 weeks ago

Why will I copy? This is RMO Delhi's problem · 7 months, 2 weeks 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. · 7 months, 2 weeks 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 :( · 7 months, 2 weeks 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. · 7 months, 2 weeks ago

rmo dehli right · 7 months, 2 weeks ago