Real fun with cyclic quadrilateral

Geometry Level 5

Let a cyclic quadrilateral \(ABCD\) inscribed in a circle with a given side length \(AB=a\) and \(AB\) subtends \(\theta \) degree angle at the center of that circle.

Maximize the area, \(A\) of quadrilateral \(ABCD\).

If it can be expressed as:

\[\displaystyle{A_{ \text{max} }=\cfrac { { a }^{ 2 } }{ p(1-\cos { \theta } ) } \times \left( q\sin { \left( \cfrac { 2\pi -\theta }{ r } \right) +\sin { \theta } } \right) }\]

for positive integers \(p,q,r\), evaluate \(p+q+r\).

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