# 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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