Rinse and repeat ....

Geometry Level 5

Suppose a cyclical quadrilateral \(ABCD\) is such that

(i) \(AB = AD = 1\),

(ii) \(CD = \cos(\angle ABC)\) and

(iii) \(\cos(\angle BAD) = -\dfrac{1}{3}.\)

The ratio of the area of \(ABCD\) to the area of the circle in which it is inscribed can be expressed as \(\dfrac{a\sqrt{b}}{c\pi}\), where \(a,c\) are positive coprime integers and \(b\) is a positive square-free integer.

Find \(a + b + c\).

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