I'm trying to construct (or find the centres/radii) of three tangent circles inscribed within the space between three tangent circles.
The end result should be as in this picture (the blue circles): https://www.ics.uci.edu/~eppstein/junkyard/tangencies/octahedron.html
Given the radii and centres of three touching circles, can one always construct this? Is it unique? - if not then I'm aiming to maximise the area of the inscribed circles.
At the moment I've got bogged down in some dense algebra of the type: $$ \begin{cases}|c_1 - c_a| = r_1+r_a \\ |c_2-c_a| = r_2+r_a \end{cases} $$ Where $c_{1,2}$ are the centres of two outer circles, and $c_a$ is the centre of a circle tangent to the other two. This says distances between centres of circlesis equal to the sum of the radii - hence are touching externally. I could (theoretically) do this for all pairs of tangent circles, and then work out the centres and radii in terms of $c_i$, $r_i$.
My question is - is there a neater way of finding the centres and radii of the inner circles without getting lost in the algebra? Or is there a nice geometric construction (which I could then use to implement an algorithm to find the centres/radii)?
Thanks in advance! :)