Three circles are drawn, so that each circle is externally tangent to the other two circles. Each circle has a radius of 2. A triangle is then constructed, such that each side of the triangle is tangent to two circles, as shown below. Find the perimeter of the triangle.
Let A = lower-left-hand vertex of the triangle.
Let B = top vertex of the triangle.
Let P = center of the lower-left-hand circle.
Let Q = center of the top circle.
Let X = point of tangency of AB with circle(P).
Let y = point of tangency of AB with circle(Q).
[The left diagonal will be A-X-Y-B.]
Since each circle has radius 2:
PQ = 4
PX = 2
QY = 2
XY = 4
Triangle(APX) is a 30o - 600 - 90o triangle with AP the hypotenuse.
Since PX =2, AP = 4, and AX = 2·sqrt(3).
YB = AX = 2·sqrt(3).
AB = AX + XY + YB = 2·sqrt(3) + 4 + 2·sqrt(3) = 4 + 4·sqrt(3)