D is the center of the larger circle of radius 4,
E is the center of the middle circle of radius R,
F is the center of the smallest circle of radius 2
Connects the radii of the three circles to form line segment DEF which has length 6+2r
THe tangent that is common to all three circles (on their tops) is line segment ABC
Next, draws the radius of each circle that is perpendicular to line segment DEF.
Then AD = 4-r, BC = R, and FC = R-2
Next, draws the line that is parallel to line segment DEF that passes through B.
The intersection of this line and AD is M. and the intersection of this line and CF is N
Note that line segment MBC is perpendicular to AD and CF .
Angles ABM and CBN are congruent vertical angles.
Angles AMB and CNB are congruent right angles.
So then triangle AMB and triangle NCB are similar right triangles by AA
By similar triangles,
(r-2)/(4-r) = (2+r)/(4+r)
(r-2)(4+r) = (2+r)(4-r)
4r + r^2 - 8 - 2r = 8 - 2r + 4r - r^2
r^2 + 2r - 8 = 8 + 2r - r^2
2r cancels; then moves everything from
right to left:
2r^2 - 16 = 0
r^2 - 8 = 0
r^2 = 8
r = 2 * sqrt(2)
The diagram has been uploaded to the RESOURCES section under filename