Three circles are drawn inside another circle such that each of the four circles is tangent to the other three circles. Two circles with radii a and b, where ab, have their centers on the diameter of the largest circle.

(a) What is the radius of the largest circle in terms of a and b?
(b) For what natural numbers a and b will the radius of the smallest circle be closest to one?

I'm trying to use Descartes' Theorem to solve it. http://en.wikipedia.org/wiki/Descartes'_theorem

Question

Three circles are drawn inside another circle such that each of the four circles is tangent to the other three circles. Two circles with radii a and b, where a>b, have their centers on the diameter of the largest circle.

(a) What is the radius of the largest circle in terms of a and b?
(b) For what natural numbers a and b will the radius of the smallest circle be closest to one?

I'm trying to use Descartes' Theorem to solve it. http://en.wikipedia.org/wiki/Descartes'_theorem

anonymous · Answer

(a) is a+b

sasogeek · Answer

do u have a diagram?

anonymous · Answer

No, this one does not have a diagram, but they are referred to as "kissing circles"

sasogeek · Answer

well twice the radius of a circle is it's diameter and since both circles touch the bigger one and their centers lie on the diameter of the bigger one. 2a +2b = diameter of larger circle, divided by 2 will give the radius... which will end up with a+b so you're right on the first one... i'm not taking a look at the second part of the question

sasogeek · Answer

that looks complex for my level, I'll see if someone else can help u on that one

anonymous · Answer

Yeah, I figured that (a) is a+b. The second part is utterly confusion, but thank you very much! You have been extremely helpful.

anonymous · Answer

ok, just brainstorming here. but would this be a visual of what they're trying to say...?

anonymous · Answer

|dw:1330496529420:dw|

sasogeek · Answer

yeah something like that :)

anonymous · Answer

I'm assuming that's what they're talking about.

sasogeek · Answer

according to the question, what natural numbers will make c close to 1. If a increases, b reduces and yet still c will reduce. but there should be a relationship between a b and c as well as the diameter of the largest circle such that we can tell the size of the circles relative to their radius and hence find c or b or a. this is what I'm thinking, i have no knowledge of this though..... educated guess. |dw:1330496767233:dw|

sasogeek · Answer

ummm i meant what natural numbers of a and b ***

anonymous · Answer

The Descartes Circle Theorem provides a relationship between all four radii. In these formulas (on the Wikipedia page), k=1/radius. http://en.wikipedia.org/wiki/Descartes'_theorem

anonymous · Answer

Any progress? What are your thoughts. This problem has me completely stumped.

hero · Answer

Good luck with that.

hero · Answer

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