All Three Concurrency

Geometry Level 5

Triangle $$ABC$$ is drawn with $$\angle B=90^{\circ}$$ and $$AC=1$$.

An angle bisector is drawn from $$A$$ hitting $$BC$$ at $$D$$. An altitude is drawn from $$B$$ hitting $$AC$$ at $$E$$. Finally, a median is drawn from $$C$$ hitting $$AB$$ at $$F$$.

Given that $$AD$$, $$BE$$, and $$CF$$ are concurrent, then the area of triangle $$DEF$$ can be represented by $\dfrac{1}{\sqrt{a+b\sqrt{c}}}$ where $$a,b,c$$ are integers with $$c$$ square-free. Find $$a+b+c$$.

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