Distances in a Right Triangle

Geometry Level 5

$$ABC$$ is a right triangle with $$\angle ABC = 90^\circ$$ with $$AB = 30\sqrt{3}$$ and $$BC = 30$$. $$D$$ is a point on segment $$BC$$ such that $$AD$$ is the median. $$E$$ is a point on segment $$AC$$ such that $$BE$$ is perpendicular to $$AC$$. $$AD$$ and $$BE$$ intersect at $$F$$. If $$EF = \frac{a\sqrt{b}}{c}$$, where $$a$$ and $$c$$ are positive, coprime integers and $$b$$ is not divisible by the square of any prime, what is the value of $$a + b + c$$?

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