The Star Medallion

Geometry Level 3

A pentagram is inscribed inside a circle with radius \(r\) as shown in the picture above. The formula that puts the area of the pentagram \(S\) in terms of the circle's radius \(r\) can be expressed as follows:

\[ \large S = \dfrac{a \sqrt{10a-b\sqrt a}}c r^2 \; , \]

where \(a,b\) and \(c\) are positive integers with \(a\) square-free and \(a,c\) coprime.

Find the value of \(a+b+c\).


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