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Given: Five circles have been inscribed in an angle (their centers are contained in the angle bisector). Adjacent circles are tangent. Express the radius of the middle circle in terms of the radii of the smallest and largest circles.

I can't seem to shake off the sine ratio (of half the given angle) in my relations. It seems that the radius of the second smallest circle is expressible in terms of the first (and likewise, the fourth in terms of the fifth). I'm having trouble putting it all together though.

Any help--even hints--would be a great help.

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up vote 5 down vote accepted

The ratio of consecutive radii is constant (by a similarity argument). The middle radius is therefore the geometrical mean of the smallest and largest radius.

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Perfect. Thank you! –  sasha Dec 12 '11 at 19:29
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