# 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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