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Is it true for infinite number of m, more than four, there exist m circles internally tangent or external tangent or combination of both each others(in this problem, i mean a circle must be tangent to all other circles, every two pair of circle tangent at a different point)? please prove this or disprove this.

Rephrase: Is it true that there exist (m>4) circles tangent to each other at different points?

i never got this because i never ever tried to generalize any geomertic theorem before.

Sorry for missing a piece of important but must for sure to edit it. Thanks in advance for answer it. This haven't answered for a day, can someone help?

Ignore the inner red circle, this picture is an example that fulfill the requirement in this problem:

enter image description here

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It's very hard to understand what your question is. Could you rephrase it? – Lubin Apr 29 '12 at 17:33
@Lubin - You see how the circle touches in my picture? – Victor Apr 29 '12 at 19:00
Are you asking whether there exists a set of $m$ circles which are all tangent to each other, for $m > 4$? The answer is no but I can only think of a proof using circle inversion. – Rahul Apr 29 '12 at 19:08
@RahulNarain - Yes, but the circles are tangent to each other at a different point. – Victor Apr 29 '12 at 19:16
@RahulNarain - if you know a proof, please answer. – Victor Apr 29 '12 at 19:20
up vote 3 down vote accepted

Choose arbitrarily any two of the circles, and perform inversion about their point of tangency. The point of tangency goes to infinity, and the two circles become parallel lines. Since the other circles do not pass through the chosen point, they remain circles that now must be tangent to these two lines and to each other.

For a circle to be tangent to both lines, it must lie between the two parallel lines and be equal in diameter to the distance between them. This reduces the possibilities of the other circles to one degree of freedom, and you can only fit two of them adjacent to each other before being unable to make any more that are tangent to all the previous ones.

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Consider four circles, all mutually tangent at different points. Invert around one of those tangent points, and the two circles tangent at that point become parallel straight lines; the other two circles become circles that are tangent to each other and to both of the straight lines. The only way that can occur is with this configuration

enter image description here

in which it is clear that no other circle can be tangent to both straight lines and both circles.

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