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May 2nd, 2007, 06:47 PM   #1
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Impossible circle problem

Hey guys

I'm having lots of trouble solving this impossible double circle problem. Can one of you please help me out? This is it:
there are 2 circles of different radii, 2.1 and 5.4, and 2 lines are tangent to them both as shown in this picture:



What is the distance between the 2 centers? Thanks alot!
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May 2nd, 2007, 08:06 PM   #2
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The result will be function of the angle between the two lines. We call it p.
Since the line joining the centers of the circles is the bisector of the aforementioned angle, you can use Thales Theorem to show that r/r'=a/a', where r,r' are the radii of the circles, and a,a' the distances from the centers of these circles to the intersection of the two lines. The distance between the two centers is given by a(r'/r+1). Now you know that r/a=sin(p/2), this a=r/sin(p/2) and the above distance equals (r+r')/sin(p/2). This corresponds to the idea that when the angle p is small (near 0), the distance will be large; when the angle is maximal (p=Pi), the two circles will have one contact point and the distance between their two centers will be exactly r+r'.
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May 3rd, 2007, 11:13 AM   #3
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Hmmm... I see that the distance is (r+r')/sin(p/2), but given just the 2 radii is it possible to find the angle p? I.e. with only those 2 lengths, can it be solved? Thanks!
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May 3rd, 2007, 03:48 PM   #4
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We have just proved that the distance between the two centers depends upon the angle between the two lines. How could you now expect the problem to be solved only with respect to the radii of the circles ?
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May 4th, 2007, 07:31 AM   #5
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Quote:
Originally Posted by g-unitcircle
Hmmm... I see that the distance is (r+r')/sin(p/2), but given just the 2 radii is it possible to find the angle p? I.e. with only those 2 lengths, can it be solved? Thanks!
Take two cups (of different sizes) and put them on the table. Now take two pencils or two rulers and make them tangent to the cups, so that when you look from above, you see your picture. Does it matter what the distance between the cups is, or can you make the pencils touch both cups, no matter what the distance is?

Sure, if the cups are too far apart, the pencils will be too short, but imagine you have very long pencils. Also, if the cups are too close, you won't be able to fit the pencils between them. Again, imagine that the pencils you have are very thin. Aside from these special cases, you can complete the picture no matter what the distance between the cups is.

So your original question does not have a unique answer. It truly is impossible.
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