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For some tasks to prove I need to know if there always exists a circle which is tangent to three given circles but contains none of the three given circles. It is ensured to have middle radius smaller than left and right radius. I know there is a problem named Apollonius' Problem which is closely related to it. I searched a bit about this on internet. But couldn't find straight answer to my question. That's why I am asking here. It'd be great if any one could provide the link of proof in favor of or contrary to the above statement.

Md. Taufique Hussain
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    if you have three circles with centers on a straight line, the outer two have the same radius but the middle one has a larger radius, you cannot do it: all six mutual tangent circles contain two or three of the circles. – Will Jagy Mar 16 '14 at 20:29
  • I see. There really are the full eight solutions. The final two have just the central circle inside. – Will Jagy Mar 16 '14 at 20:44
  • @Will, oh, I forgot to mention something. It is ensured to have middle radius smaller than left and right radius. Is it possible then in all cases? – Md. Taufique Hussain Mar 17 '14 at 03:45

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Not my best artwork, anyway, three original circles in deep purple with shading; two light blue mutual tangent circles, two light green, four orange. Should anyone have trouble seeing which colors go where, both green and blue are left-right symmetric, all three circles are inside the light blue circles/arcs, only the large central purple circle is inside the two green circles.

enter image description here

Will Jagy
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  • Is that link promoting "deep purple" music? Then maybe I should flag for spam... :) – TMM Mar 16 '14 at 22:03
  • It's not spam if it's good...right? – qwr Mar 16 '14 at 22:04
  • @TMM, you're missing the point: it's subliminal – Will Jagy Mar 16 '14 at 22:06
  • For it to be subliminal you really have to hide it better... Now it's just spam in plain sight! – TMM Mar 16 '14 at 22:08
  • @TMM Yeah, the trouble is it changes the characters in the link to blue. Quite liminal. Anyway, first i was going to write dark purple (note the colors here are not such good approximations to the original colors), then I thought no, deep purple, then i looked for youtube songs. – Will Jagy Mar 16 '14 at 22:13
  • "Should anyone have trouble seeing which colors go where, ..." ...and "deep purple" goes to YouTube :) – TMM Mar 16 '14 at 22:46
  • @TMM, "with shading" goes somewhere slightly different on youtube. I really thought they had more songs that stuck in my mind, I guess I am remembering parts of songs by quite a number of different bands. Some commonality, though, White Room and Sunshine of Your Love were by Cream, and the bands knew each other. – Will Jagy Mar 16 '14 at 22:59
  • What if both side circles have larger radius than the middle one? – Md. Taufique Hussain Mar 17 '14 at 03:49