Outside Fun

Geometry Level 5

The three blue circles with centers \(X, Y\) and \(Z\) are each tangent to two of the lines making up the triangle \(ABC\) and each of them has twice the area of the incircle of the triangle \(ABC\). If the area of the triangle \(ABC\) is 1, find the area of the triangle \(XYZ\) and express it as \(a+b\sqrt{c}\), where \(a,b\) and \(c\) are positive integers with \(c \neq 1\) square-free. Submit your answer as \(a+b+c\).

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