A convenient transformation, when you are dealing with circles, is inversion.
Inversion takes circles and straight lines to circles and straight lines. Moreover, if a circle passes through the center of inversion then it turns into a line. Here we have circles tangent to each other. So, if we use as center the tangency point of two of the circles, they both will turn into lines. Even more, since inversion preserves angles, these two lines will be parallel.
Take the tangency point, $P$, of two of the given circles, $c_1$ and $c_2$, and apply inversion with center $P$ and radius, say $1$.
(Actually it might be better to use a radius, if possible, such that the circles get intersected at two points. That way you know that those two points are not going to move during inversion.)
The two circles $c_1$ and $c_2$ are going to become parallel lines, $L_1$ and $L_2$, so the center of the circle you are looking for lies in the middle line, $L_3$. Notice that $L_1$ and $L_2$ are easy to construct since they pass through the intersection points of a circle with center $P$ and radius $1$ and each circle $c_1$ and $c_2$. In the worst case you can always invert two or three points of the circle, as needed, and the inverse line or circle will pass through those inverted points.
The other circle $c_3$ becomes a circle $C_3$ in between the two lines because tangency (i.e. angles) is preserved. So, the tangency points of the inversion $C_4$, of the circle to be constructed to that circle $C_3$ are the intersection of the middle line $L_3$ with that third circle, $C_3$.