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Positions of points in intersecting circles (DIFFICULT)

Hi! here is the task I was talking about in my introductory thread I just posted a few minutes ago. I got this task yesterday and I have no idea how to even start with it. Here is an image of the task. I read it 100 times but I still don't know what to do! (Crying)

Attachment 26817

Re: Positions of points in intersecting circles (DIFFICULT)

You should post this here: Trigonometry

To get you started, try reading this: Intersecting Circles

Re: Positions of points in intersecting circles (DIFFICULT)

Hi NatalieSv94! :)

The triangle OPA is an isosceles triangle.

Furthermore the triangle OP'A is also an isosceles triangle.

Moreover this second triangle has a common angle with the first, meaning that these two triangles are *similar*.

Can you say anything about what that means for OP'?

Re: Positions of points in intersecting circles (DIFFICULT)

I dunno (Worried). OP'/AO=AP'/PO=AO/AP ? Is that right? And then, what am I supposed to do?

Re: Positions of points in intersecting circles (DIFFICULT)

That is good! :D

So suppose r=1 and OP=2, as your problem states.

Then AO, which is the radius, is also 1.

What is OP' then?

Re: Positions of points in intersecting circles (DIFFICULT)

It must be 1/2! Right? :D

Thank you so much ;)

Re: Positions of points in intersecting circles (DIFFICULT)

Yep! It is 1/2!

Can you find the general statement?

Re: Positions of points in intersecting circles (DIFFICULT)

Re: Positions of points in intersecting circles (DIFFICULT)

Re: Positions of points in intersecting circles (DIFFICULT)

Right!

That is for r=1.

So the next challenge is, what if r=2?

Re: Positions of points in intersecting circles (DIFFICULT)

I'll find it! OMG! I'll find it! it's OP'=r^{2}/OP !!!! Is that right?

Re: Positions of points in intersecting circles (DIFFICULT)

Yep, that is right! (Wait)

Re: Positions of points in intersecting circles (DIFFICULT)

OMG! Thank you sooooooooooooooooooooo much! :) You're a great teacher you know!

In the last part when it says Discuss the scope and/or limitations of the general statement. I know that OP' approaches infinity as r approaches infinity and OP' approaches 0 as OP approaches infinity. Is that all? Or is there any situation in which OP' is not defined or the general statement is not working? (I know that it's not defined if OP is 0, but ...)

Re: Positions of points in intersecting circles (DIFFICULT)

Thanks! (Blush)

The only real limitation is that OP may not be zero, so the point P may not be chosen to coincide with the point O.

EDIT: Hold on! See below.

Re: Positions of points in intersecting circles (DIFFICULT)