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Suppose that we have two distinct circles and a line on a plane and that the distance between the centers of the circles is bigger than the sum of their radiuses. Also, suppose that the two circles exist in the same half plane divided by the line.

enter image description here

Question : If we consider a circle which is tangent both to the two circles and to the line, then what is the max of the number of such circles?

I have the following conjecture.

Conjecture : The max of the number of such circles is $8$.

We can see that $8$ is possible.

enter image description here

However, I'm facing difficulty in proving the conjecture (but it is just a conjecture). Can anyone help?

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  • $\begingroup$ Are the two circles allowed to be tangent to the line? (I don't know if that changes the answer.) $\endgroup$
    – LarsH
    Nov 25 '14 at 14:48
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    $\begingroup$ @LarsH: Yes. (The wiki page shows that that does not change the answer, I think.) $\endgroup$
    – mathlove
    Nov 27 '14 at 10:37
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This is the Apollonius third problem with a degenerate circle.

The paragraph algebraic solutions of the Wikipedia article clearly shows that there are at most eight ($2^3$) solutions.

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    $\begingroup$ Thank you for the information! Very interesting! $\endgroup$
    – mathlove
    Nov 25 '14 at 12:44

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