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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