Two unit circles are completely enclosed by a

*ring* of four identical circles. Find the minimum radius of the surrounding circles.

Note: A *ring* of circles is a set where each is externally tangent to exactly two others and they enclose a single region.

With two of the ring circles situated side by side, a small circle of radius r is nestled between them such that the top of their circumferences our co-linear.

If we draw a triangle between a ring circle center, the two ring circles' tangent point

and the center of the small circle,

**we see this **is a (1-r, 1, 1+r) right triangle.

(1-r)^2 + 1^2 = (1+r)^2

**r=1/4**

*Edited on ***August 30, 2024, 10:50 am**