Three tangent circles calculator

Area between three tangent circles
We define:  a = r_{2} + r_{3}  b = r_{1} + r_{3}  c = r_{1} + r_{2} 
The area of triangle ABC is:  
The sectors area are: 
And the area between the 3 tangent circles (green area) is:
A = A_{T} − A_{A} − A_{B} − A_{C}
The angles of the triangle ABC can be found by cosine law:
The green area circumference is:  P = α r_{1} + β r_{2} + γ r_{3} 
The radii of the four tangent circles are related to each other according to Descartes circle theorem:
The plus sign means externally tangent circle like circles r_{1} , r_{2} , r_{3} and r_{4} and the minus sign is for internally tangent circle like circle r_{5} in the drawing in the top.

And the curvature of the circles k_{4} and k_{5} which are called the Soddy circles are:
If circle r_{1} is a straight line then r_{1} = ∞ and the curvature is k_{1} = 1 / r_{1} = 0 The curvature of the two red Soddy circles are simply: 
Area between three tangent circles example
First, we will find the value of d
The area of the trapezoid BCDE is:

The circumference of the green area is: 
NOTE: in all the calculations we assumed that r_{2} > r_{3}