Maximizing the area!

Geometry Level 5

If \(AB\) is a fixed line segment with the length of \(AB\) being \(1\) unit, find the triangle \(ABC\), which has a maximum area among those which satisfy \(\angle AIO = \dfrac{\pi}2\), where \(I\) is the incentre of \(\Delta ABC\), and \(O\) is its circumcentre. If the maximum area of that triangle can be represented as:

\[\large{\dfrac{\sqrt{A+B\sqrt{C}}}{D}}\]

where \(A,B,C,D\) are positive integers, and \(C\) has no square factor. Submit the minimum value of \(A+B+C+D\) as your answer.

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