Given two touching circles, find position of a third circle of known radius so that it touches them To give an idea of what my end goal looks like visually:

You start with one circle. You add a second circle, making it touch the first circle at some point. For each successive circle (for which we know the radius) to be added, there should be exactly two ways for it to touch the previous two (for which we know the coordinates of their centers and their radii). Is there a formula that expresses these solutions?

We have two circles, with radii $r_1$ and $r_2$ and centres $(x_1,y_1)$ and $(x_2 ,y_2)$ respectively. These two circles touch, so we can say that the distance between the two centres is $$\sqrt{(x_1-x_2)^2+(y_1-y_2)^2} =r_1+r_2$$
We have a third circle with known radius $r_3$ and we want to find its possible centres $(x_3,y_3)$.
We know that the distance between the centre of circle $1$ and $3$ will be $$\sqrt{(x_1-x_3)^2+(y_1-y_3)^2} = r_1+r_3$$ and the distance between the centre of circle $2$ and $3$ will be $$\sqrt{(x_2-x_3)^2+(y_2-y_3)^2} = r_2+r_3$$
We can see this in the below diagram:

We can the use these two equations to solve for $(x_3,y_3)$
\begin{align}\sqrt{(x_1-x_3)^2+(y_1-y_3)^2} &= r_1+r_3\tag{1}\\
\sqrt{(x_2-x_3)^2+(y_2-y_3)^2} &= r_2+r_3\tag{2}\end{align}
as we know the values of all the other variables, $x_1,y_1,r_1,x_2,y_2,r_2,r_3$

Example:
We have a circle centre $(1,1)$, radius $2$, and a circle centre $(4,5)$, radius $3$. We want to add another circle of radius $1$. 
We input all these values into the formulae above:
\begin{align}\sqrt{(1-x_3)^2+(1-y_3)^2} &= 3\tag{1}\\
\sqrt{(4-x_3)^2+(5-y_3)^2} &= 4\tag{2}\end{align}
We can solve these for $(x_3,y_3)$ to find that the two possible centres are $$(4,1)\text{ and }\left(\frac 4{25},\frac {97}{25}\right)$$
I'll leave it to you to plot these $4$ circles and see that it works!
