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Why is Descartes' theorem failing here?

I was doing a simple question :

Three circles of equal radii, equal to $1$ unit are touching each other. Find the radius of circle circumscribing the three circles.

My answer: Radius of circumcircle of triangle formed by joining the centre of the three smaller circles is $$2(√3/2)(2/3) = 2/√3$$

so, radius of circle circumscribing the three circle will be $$(2/√3) + 1$$ which is correct as per the book.

Then I thought of applying Descartes' theorem, which gave me $$1/(3+2√3)$$ which is wrong.

Am I committing any mistake? Why not both the answers are same? Please help.

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1 Answer

Descartes' theorem gives the curvature of the outer circle as $3\pm 2\sqrt{3}$, with a negative value indicating a circle that circumscribes the others. So in this case, the radius is $-1/(3-2\sqrt 3)$, which turns out to be equal to the answer you got by the first method.

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Got my mistake, thank you sir. –  Hyperbola Jul 21 '12 at 8:56
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