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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
2 weeks, 4 days 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. · 5 days, 15 hours ago

Very good solution. Very much clear also. I understood it now. · 3 days, 22 hours 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?? · 3 days, 19 hours 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? · 1 day, 16 hours ago

Please share these resources with me also.

My email id : harshus99@gmail.com · 23 hours ago

and also give the INMO paper to this email.Did u write RMO this year · 1 day ago

Sometimes co-exter or pre-college maths.Can u give the pdf of the books to this email : raiayush234@gmail.com · 1 day ago

This is there in IF Sharygin, plane geometry. · 2 weeks, 3 days ago

A problem copied from IF Sharygin Plane geometry. This is bad. · 2 weeks, 3 days ago

Why will I copy? This is RMO Delhi's problem · 2 weeks, 2 days 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. · 2 weeks, 1 day 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 :( · 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. · 1 week, 6 days ago

rmo dehli right · 2 weeks, 3 days ago