Solving a nasty set of equations

[Mr.Miyagi]

Suppose I have the following set of equations
$$x=3\pi r-r-2\sin\theta$$
$$y=r-2\cos\theta$$
$$\tan\theta=y/x$$
with r as a constant

How would I go about finding out if an analytical solution exists? There must be a solution, as I distilled the equations from a fairly straightforward geometrical problem that I conjured up.
I just can't seem to find it.

The geometrical problem is basically finding the length of the inner tangent of two circles, between the points where the tangent and the circles touch plus parts of the circles. It's length would be $$\sqrt{x^2+y^2}$$ and the parts of the circle have a length $$2\theta r$$. I hope this makes sense.

 

[tiny-tim]

Hi Mr.Miyag!

(have a theta: θ )

Rewrite sinθ and cosθ in terms of tanθ,

then convert them to an expression in y/x, and solve.

 

[Mr.Miyagi]

Hi:)

I still don't see how I can solve for all three variables.
$$\sin\theta=\tan\theta\cos\theta=\cos\theta\frac{y}{x}$$
$$\cos\theta=\tan\theta\frac{\cos^2\theta}{\sin\theta}=\tan\theta\cot\theta\cos\theta=\cot\theta\cos\theta\frac{y}{x}$$
then
$$x=(3\pi-1)r-2\cos\theta\frac{y}{x}$$
$$y=r- 2\cot\theta\cos\theta\frac{y}{x}$$
I don't see a substitution that simplifies things.

 

[tiny-tim]

Hi Mr.Miyagi!

(what happened to that θ i gave you? )

… I don't see a substitution that simplifies things.

You need to learn your trigonometric identities …

in particular, sec²θ = tan²θ + 1

 

[Mr.Miyagi]

(What's up with the theta?)

Ah, I overlooked the identity. I can express x as a function of y and vice versa, with the trig identity now. But it doesn't seem to help me in finding θ as a function of r.(I shouldn't have said r is a constant. It is just a variable).
I feel I'm overlooking something really obvious, but I can't seem to figure it out.

 

[flatmaster]

I don't quite understand the geometry you are trying to explain. How are these circles defined again? Are they concentric.

Also, if your x,y,r,and theta are both the coordinates, you could use any of the equations that do {x,y}-->{r,theta} and visa versa.

For example,

x = r Cos[theta]
y = r Sin[theta]

 

[Mr.Miyagi]

I apologize for the poor choice of variable names. I've made a drawing to explain what I was talking about. It's in the attachment.
The line GH would be our x, DH would be our y, the radius of the circles is our r and the angles denoted apha are our theta.
It is also known that the line BC has length $$(3\pi-1)r$$
Now I'd like to know the length of the line DG and the angle of alpha.