I wonder if there is research into generalizing Apollonius's problem to create tangent circles for three conic sections at the same time. I know that Apollonius' basic problem was already faced by many people, as this issue is enormously more complex. I know that it is illogical to ask the question of how to create, It will require one or more books to complete the various types of issue, but please provide references, links, and research on this, if any.
I would be very happy if some cases were addressed in the solutions
Note: I do not know whether the problem can be constructed essentially using a ruler and compass, so if it is not solvable then it is permissible to use conic sections
As a starting point, I'd look at an algebraic approach. Let $A,B,C$ be symmetric $\mathbb R^{3\times 3}$ matrices representing the three given conics, and let your solution circle be described by the matrix
$$M=\begin{pmatrix}a&0&b\\0&a&c\\b&c&d\end{pmatrix}$$
based on the four indeterminates $a,b,c,d$ which parametrize the circle (or a circle degenerated to a finite line and the line at infinity).
A linear combination of conic sections, such as $M+\lambda A$ in general has three special values for which the resulting combination is degenerate, i.e. has determinant zero. These correspond to pairs of lines through the four points of intersection, which may be complex. If the two conics touch one another, then two of the points of intersection coincide, which means two of the solutions $\lambda$ coincide as well.
So if you treat $\det(M+\lambda A)$ as a cubic polynomial in $\lambda$, and then compute its discriminant, setting that discriminant to zero gives you a (necessary and I believe in general also sufficient) condition for conics $A$ and $M$ touching one another. This condition will depend on $a,b,c,d$. Do the same for conic $B$ and $C$ instead of $A$, and you get three homogeneous conditions on the indeterminates of your circle. If you can exclude the case where the circle degenerates into a line, then you can assume $a=1$ as a fourth equation, and feed all four of them to a computer algebra system in the hope of getting back some numbers for $b,c,d$. Or use some tool that can solve systems of homogeneous equations without artificially pinning one of the variables.
However, the equations you get from each discriminant is of combined degree $6$ in the indeterminates of your circle. I'm still waiting on some concrete results from an example testcase I have in my Sage notebook, where I started from three random symmetric matrices with integral entries. I wouldn't be surprised if that degree propagates to the minimal polynomial of the resulting algebraic numbers, probably made considerably worse by the fact that three such polynomials get combined. I would expect this to expresses itself in an unsolvable Galois group of the minimal polynomial. If that were the case, then no radical expression could be given for these numbers, which rules out any construction even if you allow intersecting conic sections as one of the operations. Not sure how much longer that computation will take; it's been several hours already. So at the moment the insolvable groups are merely a conjecture.
Update, some hours later: I still don't have a solution to my randomized example test case. But after fixing $a=1$ and eliminating $c$ and $d$ using resultants, I got the polynomial for $b$ factorized. It had two factors, one of degree 176, the other of degree 550. On the one hand this may help us understand why computing a solution is taking so long. On the other hand, I would be extremely surprised if this monster of an equation were to allow for any solution by radicals, even though computing Galois groups for polynomial of this degree is beyond Sage's capabilities. While the size of these polynomials, and the duration of my computation, make the algebraic approach appear infeasible for many practical application, I believe that the complexity is pretty much inherent in the problem itself, at least if you are looking for exact solutions and not just some numerical approximation techniques.