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.