# How does one find a circle which intersects only once with three other circles?

Say I have the equations of three different circles on a plane. How would I proceed to create a fourth circle, which intersects only once with each of these known circles? Would that be possible at all?

Edit: The equations for the circles here are dynamic and my goal is to find a generalised method, however the equations I'm currently working with are: $$(x+1)^2+y^2=(\sqrt 17 - \sqrt 5 )^2$$ $$x^2+(y+1)^2=(\sqrt 13 - \sqrt 5 )^2$$ $$x^2+(y-1)^2=(3 - \sqrt 5 )^2$$ Sorry they look so messy. The circles are not assumed to be tangent, however the fourth one I'm looking for is tangent to all three.The fourth circle should not enclose any of the three other circles. The three circles do not share a center.

As stated by @Rahul in the comments, this is Apollonius' Problem. There's one circle which externally touches each circle and it can be found using the set of equations shown on the wiki page. Thank you Rahul.

• This is known as Apollonius's problem. There are typically eight choices for the fourth circle, depending on which of the given three circles, if any, the fourth circle should enclose. – Rahul Apr 23 at 12:21
• If your three circles are concentric, with differing radii, I don't think it is possible. – kimchi lover Apr 23 at 12:22
• Are you assuming the original 3 circles are tangent? If so, please edit your question accordingly. – kimchi lover Apr 23 at 12:23
• The algebraic solution on the Wikipedia page isn't just about the tangent circle that all the others are inside. It is equally good for any of the other seven circles. For your particular case, try $s_1=s_2=s_3=-1.$ – David K Apr 23 at 13:16
• @Jashani, the radii of two first circles are too large, so they cover partially each other. If you think they are dynamic, put a letter instead of a concrete value for the radius. – user376343 Apr 25 at 10:57