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There is one circle with radius $1$. There is another circle with radius $2$. They are tangent to each other and touch each other at point $c$. A line through $c$ splits the area formed by the two circles into two parts. The ratio of the two parts is $1:2$. In what ratio does the line split the area of the smaller circle (the circle with radius $1$)?

enter image description here This was one of the questions in last year's AMC. I have no idea how to solve it. I thought that you could try and work out the of the two parts of the smaller circle and figure out what the ratio was between them. I tried that but failed. Thanks for any help.

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  • $\begingroup$ If anyone has a picture for this problem, could you please edit my question and upload the picture? $\endgroup$ – Joao Sep 17 '14 at 6:39
  • $\begingroup$ Are the two circles and dividing chords similar figures? $\endgroup$ – Henry Sep 17 '14 at 6:56
  • $\begingroup$ @Henry Well, two circles are always similar right? I don't know about the chords. $\endgroup$ – Joao Sep 17 '14 at 7:00
  • $\begingroup$ I would have thought the chords had the same angle with the tangent, and so the same angles with the radii. $\endgroup$ – Henry Sep 17 '14 at 7:03
  • $\begingroup$ @Henry I don't know... sorry I don't know much about geometry. $\endgroup$ – Joao Sep 17 '14 at 7:09
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enter image description here

Regions P and S are similar, as are regions Q and R. And the area of the right-hand circle is four times the area of the left-hand circle. So $P = 4S$ and $Q = 4R$.

And we are told that $Q+S = 2(P+R)$.

Substituting, $4R+S = 2(4S+R)$. You can take it from there.

Updated to add: Riffing on André's comment (now deleted), this doesn't just work for circles. For instance, it works for Australias too:

enter image description here

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  • $\begingroup$ Yes I can! Thanks its all clear now. $\endgroup$ – Joao Sep 17 '14 at 7:17

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