Circles

Given that ω1\omega_1 and ω2\omega_2 are two circles which intersects at points XX and YY. Let PP be an arbitrary points on ω1\omega_1. Suppose that the lines PXPX and PYPY meet ω2\omega_2 again at points AA and BB, respectively.

Prove that the circumcircles of all triangles PABPAB have the same radius.

Note by Sayantan Saha
3 years ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Problem 1 Problem 1

We can see that as we move the point PP on the circumference of the circle[[excluding XX and Y],Y],the XPY=XP1Y\angle XPY=\angle XP_1Y remains constant.So this shows that AB=A1B1.AB=A_1B_1.Now we use extended sin rule to complete the problem.
Let the circum-radius of PAB\triangle PAB be RR and P1A1B1\triangle P_1A_1B_1 be R1.R_1.
In PAB,ABsinP=2R\triangle PAB, \frac{AB}{sin\angle P}=2R and in P1A1B1,A1B1sinP1=ABsinP=2R1.\triangle P_1A_1B_1,\frac{A_1B_1}{sin\angle P_1}=\frac{AB}{sin\angle P}=2R_1. Therefore 2R=2R1R=R1.2R=2R_1\Rightarrow\boxed {R=R_1}.Hence Proved.

Ayush G Rai - 3 years ago

Log in to reply

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

Priyanshu Mishra - 2 years, 12 months ago

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??

Ayush G Rai - 2 years, 12 months ago

Log in to reply

@Ayush G Rai 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?

Priyanshu Mishra - 2 years, 12 months ago

Log in to reply

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

Ayush G Rai - 2 years, 12 months ago

Log in to reply

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

Ayush G Rai - 2 years, 12 months ago

Log in to reply

@Ayush G Rai Yes I appeared in RMO this year from KVS region.

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra How many did u solve?

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Ayush G Rai I solved 4 and half.

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra Awesome! Send me the question paper also then

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Ayush G Rai You will get it.

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

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra ok.no probs. Can u give ur email address

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Ayush G Rai priyanshu_2feb@rediffmail.com

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra Please share these resources with me also.

My email id : harshus99@gmail.com

Harsh Shrivastava - 2 years, 12 months ago

Log in to reply

rmo dehli right

abhishek alva - 3 years ago

Log in to reply

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

Sayantan Saha - 3 years ago

Log in to reply

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

Priyanshu Mishra - 3 years ago

Log in to reply

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

Sayantan Saha - 3 years ago

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.

Priyanshu Mishra - 3 years ago

Log in to reply

@Priyanshu Mishra 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 :(

Harsh Shrivastava - 3 years ago

Log in to reply

@Harsh Shrivastava 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.

Priyanshu Mishra - 3 years ago

Log in to reply

This is there in IF Sharygin, plane geometry.

Priyanshu Mishra - 3 years ago

Log in to reply

@Harsh Shrivastava, @Ayush Rai,

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

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

yup.sure.send it to my email.

Ayush G Rai - 2 years, 11 months ago

Log in to reply

Ok no problem.

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

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra H. S. M. Coxeter and S. L. Greitzer

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Ayush G Rai Is it Geometry revisited by MAA?

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra yup

Ayush G Rai - 2 years, 11 months ago

Log in to reply

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

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra yup.i use it to learn new stuff.

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Ayush G Rai Do you have any good book regarding NUMBER THEORY?

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra David M.burton book;s

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Priyanshu Mishra can u give ur email ID

Ayush G Rai - 2 years, 11 months ago

Log in to reply

@Ayush G Rai I gave. See above.

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

Yeah I also want em all.

Harsh Shrivastava - 2 years, 11 months ago

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.

Priyanshu Mishra - 2 years, 11 months ago

Log in to reply

Thanks a lot bro.

Ayush G Rai - 2 years, 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...