The trick to solving this is choosing the correct obtuse angle to bisect; get that right and my proposed solution will give the correct answer.
As I stated, I was bisecting the angle between r0r1 and r0r2 (i.e. the angle between the 3 circles’ centres – C1-C0-C2). The problem with this is that it fails to take into account the radii of the two circles – it is only when they are equal that my original approach will correctly calculate r3. When C2 > C1 the bisected angle is too small and r3 is smaller than the true value, and vice versa when C2 < C1. The greater the relative difference, the greater the error.
My solution is to therefore use the obtuse angle between the tangents of C1 with the centre of C0 and C2 with C0, and bisect that. The first figure in my (rough and ready) diagram, shows the kind of scenario I’m interested in.
A right triangle can be constructed between the centres of C1 and C0, and a tangent between circle C1 and centre of C0. Using Pythagoras and a bit of trigonometry we can calculate the angle that C1-C0 makes with the tangent, as follows:
θᵢ = sin¯¹ (rᵢ / (r₀ + rᵢ) )
This calculation must be done for both circles C1 and C2 (i.e. for i =1 and i = 2), giving θ₁ and θ₂ respectively.
If (big) ϴ is the angle C1-C0-C2, then the bisected angle between the two tangents is simply:
θ₁,₂ = (ϴ - θ₁ - θ₂) / 2
Since this is the angle from each tangent to the bisector (the line marked B in my diagram), and we want the angle from C0-C2 to the bisector, we must add θ₂:
θ₁,₂ = (ϴ - θ₁ - θ₂) / 2 + θ₂
= (ϴ - θ₁ + θ₂) / 2
It is this value for theta that we then use in the formulas that I originally gave in my question.
How big is the difference in theta for the two methods? Well, for the circles in figure 1, line A represents the original theta and line B the method described here (around 1.5° or 0.027 radians). Not a great difference, given C1 and C2 are relatively close in size, but we want this to be right!
I have provided the second figure for information only, but it shows the kind of scenario where my first approach would have failed (the angle C1-C0-C2 is not obtuse in the position where we want to position C3, but is obtuse when we consider the tangents)
Finally, I am just an interested amateur and not a professional mathematician, so I may have broken all sorts of rules in documenting my approach to solving this problem. I’m really hoping I’ve not made a fundamental mathematical error somewhere (it does seem to work pretty well for me though)! I would love for someone to correct me if I’ve committed any cardinal sins…