You can use a special line known as the radical axis.
The radical axis is the locus of all points on the plane that have the same power with respect to the 2 circles in question, i.e. the length of the tangent to both circles from any point on this line is the same.
It is know that for any 2 non-concentric (sharing a center) that are tangent to each other, the shared tangent is the radical axis.
Thus it can be shown that for any 3 circles, the 3 tangents formed will always be concurrent. The proof is as follows:
Let $w_1$,$w_2$ and $w_3$ be the 3 circles that are tangent to each other. We can assume that none of them share a center as it would be the same as 2 circles being tangent.
We know that for any point $P$ on the radical axis of $w_1$ and $w_2$, the power of P with respect to $w_1$ and $w_2$ is the same.
We also know that for any point $P$ on the radical axis of $w_2$ and $w_3$, the power of P with respect to $w_2$ and $w_3$ is the same.
Since the 2 lines must intersect (they cannot be parallel as this would imply that either they are not tangent to each other or at least 2 of them are identical and concentric), the point where the 2 radical axes meet has the same power with respect to $w_1$,$w_2$ and $w_3$. This implies that it must also be on the radical axis of $w_1$ and $w_3$. Therefore, this point is where the 3 tangents are concurrent.