Equation of 3 circles touching each other is given, what is the equation of a circle touching all other 3?
does it matter that there are 2 circles that can touch all other 3 circles, one being between the 3 and the other having the 3 inside them? (not considering the cases were 1 or 2 touching circles are already contained in a circle)
This is the Soddy's circles problem, a special case of the Problem of Apollonius.
The first link shows you how to find the radius of either of your solution circles: if the radii of the three given circles are $r_1,r_2,r_3$ you can find the desired radius $r$ by
$$\left(\frac 1{r_1}+\frac 1{r_2}+\frac 1{r_3}+\frac 1{r}\right)^2 =2\left(\frac 1{r_1^2}+\frac 1{r_2^2}+\frac 1{r_3^2}+\frac 1{r^2}\right)$$
Solving that leads to a quadratic equation in $r$ with two solutions, showing that there are indeed two circles that solve the problem.
I'll leave it to you to find the centers of those circles.