Twisting the Problem - "Complicated Tangency of a system of circles"!

Geometry Level 4

Eight equal circles are mutually tangent in pairs and tangent externally to a unit circle as shown in the figure. If the common radius of the eight smaller circles can be expressed as:

\[\large{\dfrac{\sqrt{A-\sqrt{B}}}{C - \sqrt{D-\sqrt{E}}}}\]

for positive integers \(A, B, C, D, E\) which doesn't have any square factor. Find the minimum value of \(A+B+C+D+E\).

Here is a senior version of this problem - Complicated Tangency of a system of Circles!

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