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Three circles of equal radii have been drawn inside an equilateral triangle , of side a , such that each circle touches the other two circles as well as two sides of triangle.

Then find the radius of the circle. I tried but could not get where to start. I think we need to relate the center of circle with either incentre / circumcenter.

Will there be any relation between incentre / circumcenter with the centres of 3 circles?

How i can achieve this? Thanks in advance.

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2 Answers 2

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Hint. Consider two of those circles, and the rectangle given by 4 points: two centers of these circles, and two their tangency points on one side of the triangle. What can you tell about the proportion of the sides of the rectangle? Do you know where the two centers of the circles are placed? Does it help?

If you get something like $x+2\cdot(x\cdot\tan\frac{\pi}{6})+x=a$ then you are thinking in the right direction, I guess.

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  • $\begingroup$ Let me try again and think. $\endgroup$
    – vikiiii
    Commented Apr 22, 2012 at 3:34
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$$a=2r +2r\tan 60^{\circ} \Rightarrow a=2r(1+\sqrt 3) \Rightarrow r=\frac{a}{2(1+\sqrt 3)}$$

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