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Circle problem (2)

Radiusları Rr olan iki çevrə xarici toxunandır. Bu çevrələrə və bu çevrələrin xarici toxunanına toxunan çevrənin radiusunu tapın.

Given two circles (O1,r) & (O2,R). Find the radius of third circle (O3) .

     Solution :

Let the radius of the third circle be x.

AB^2 = (R+r)^2 - (R-r)^2 = 4Rr \Rightarrow

\Rightarrow AB=2 \cdot \sqrt{Rr}

AC^2 = (r+x)^2 - (r-x)^2 = 4rx \Rightarrow AC = 2 \cdot \sqrt{rx}

CB^2 = (R+x)^2 - (R-x)^2 = 4Rx \Rightarrow CB = 2 \cdot \sqrt{Rx}

AB = AC + CB \Rightarrow 2 \sqrt{x} \cdot ( \sqrt{R} + \sqrt{r} ) = 2 \sqrt{Rr} \Rightarrow \\ \\ \Rightarrow x= \frac{Rr}{ ( \sqrt{R} + \sqrt{r} )^2 }

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