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In figure, if the distance between two small circles is 84, what will be the distance between two large circles?

Source: Bangladesh Math Olympiad 2016 Junior Category

I can't solve this this problem. But I found that the connected lines between the centers of the 4 circles create a rhombus. The diagonal of the rhombus is 84.

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    $\begingroup$ The vertical diagonal of the rhombus is $84$. $\endgroup$ – TonyK Jan 28 at 13:36
  • $\begingroup$ @TonyK Yes, I know that. But how am I supposed to find the other diagonal of that rhombus? $\endgroup$ – Shromi Jan 28 at 13:41
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    $\begingroup$ If you know that, why didn't you say it? $\endgroup$ – TonyK Jan 28 at 13:51
  • $\begingroup$ I wrote that in my question. $\endgroup$ – Shromi Jan 28 at 13:53
  • $\begingroup$ I give up.${}{}$ $\endgroup$ – TonyK Jan 28 at 14:22
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Hint:

If the large circles each have radius $R$ and the small circles $r$ then:

  • the distance between the centres of the small circles is $84=2R-2r$ and half of this is $42=R-r$
  • the distance between the centre of a small circle and the centre of a large circle is $R+r$
  • the distance between the centres of the large circles is $2R$ and half of this is $R$
  • You have right-angled triangles so can get a second equality to solve simultaneously with $42=R-r$
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Let the radii of the large and small circles be $R$ and $r$. Then Pythagoras gives $$(R+r)^2=R^2+(R-r)^2$$ which simplifies to $$4r=R$$ We are given that $2R=84+2r$. You should now be able to work out what $2R$ must be.

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Hint:

If $r$ is the radius of large circle and $s$ of small circle then:$$r^2+(r-s)^2=(r+s)^2$$leading to $r=4s$.

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