Place a circle 'on top' of two other circles?

Question

I have two circles (their radii and position) given. I then have a third circle (only it's radius), and would like to calculate its position so it touches both other circles:

There are always two points where the third circle can be placed, but the third circle should be place so that when 'walking' from the first to the second circle, the third is on the left. So on my drawing, the first circle is the big one on the left, the second the smaller one on the right, and the third is the orange one. The arrow shows the 'walking' path.

How can I do this?

Answer 1

The first two circles have radius $R1$ and $R2$; your third circle will have radius $R3$.
The centre of the third circle is $R1+R3$ from one centre, and $R2+R3$ from the second.
If the centres are $C1,C2,C3$, then you want the cross-product $(C2-C1)\times(C3-C1)$ to be positive (I think). Certainly, the sign of the cross-product is what you need.

Answer 2

We know that, when two circles touch each other, then the distance between their centers is equal to the sum of their radii.

Let the radii of the circles with centers $A, C \ \text{and}\ E$ be $r_1,r_2$ and $R$ respectively. Then, $$EA=R+r_1, \\ EC=R+r_2$$ so that $$|EA-EC|=|r_1-r_2|,$$ which is a constant. Thus, the locus of the center of the required circle is a branch of a hyperbola with foci $A$ and $C$.

You can plot the points A and C on a coordinate system (with maybe A as origin), then derive the equation of the hyperbola.
Next, use the parametric form of the hyperbola to represent the center of the required circle and find the center by using $EC=R+r_1$. You can see this post for further information.