﻿ Three tangent circles
 Three tangent circles ▲ r1 r2 r3 Green area Green area length Triangle ABC area Angle α Angle β Angle γ Inner circle radius   r4 Outer circle radius   r5
 Area between three tangent circles ▲
 We define: a = r2 + r3 b = r1 + r3 c = r1 + r2
 The area of triangle ABC is: The sectors area are: And the area between the 3 tangent circles (green area) is:            A = AT − AA − AB − AC
The angles of the triangle ABC can be found by cosine law: The green area circumference is: P = α r1 + β r2 + γ r3
The radii of the four tangent circles are related to each other according to Descartes circle theorem: If we define the curvature of the  nth  circle as: The plus sign means externally tangent circle like circles   r1 , r2 , r3 and r4   and the minus sign is for internally tangent circle like circle   r5 in the drawing in the top.
 Then the Descartes circle theorem is: ( k1 + k2 + k3 + k4 ) 2 = 2 ( k12 + k22 + k32 + k42 )
And the curvature of the circles k4 and k5 which are called the Soddy circles are:  If circle r1 is a straight line then   r1 = ∞ and the curvature is   k1 = 1 / r1 = 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 trapazoid   BCDE   is:   The circumference of the green area is:  NOTE: in all the calculations we assumed that   r2 > r3